Area and Perimeter — Extra Practice

Competitive Mathematics · Geometry

9 Part21 worked examples466 practice questions

NoteThis is the Extra Practice set: lots of questions. To learn the methods first, read the main chapter of the same topic (the one without “Extra Practice” in its name), then practise here.

To tile your room’s floor you need the space (area); to put a border along the wall you need the edge (perimeter). Perimeter = the length all round (cm, m); area = the space inside (sq cm, sq m). In this chapter we learn to find both for every figure — and then see how they change when sides change or a path is built.

∑ Formula

आयत : A = l b, P = 2 (l + b); वृत्त : A = π r^{2}, C = 2 π r; त्रिभुज : A = \frac{1}{2} bh

The three most important formulas

Triangles

Half base times height

Heron’s formula, right-angled, isosceles, equilateral, similar triangles and limits on sides.

Types and formulas

A formula for each kind

First identify the kind of triangle.

◭ Diagram

Triangle: area

= \frac{1}{2} \times b \times h

; perimeter

=

sum of the three sides

Triangle	Area	Perimeter
Any, sides $a, b, c$	$s (s - a) (s - b) (s - c)$ , $s = \frac{a + b + c}{2}$	$a + b + c$
Right-angled (perp. $p$ , base $b$ , hyp. $h$ )	$\frac{1}{2} p b$ , $h^{2} = p^{2} + b^{2}$	$p + b + h$
Equilateral (side $a$ )	$\frac{3}{4} a^{2}$ ; ऊँचाई $\frac{3}{2} a$	$3 a$
Isosceles (equal $a$ , base $b$ )	$\frac{b}{4} 4 a^{2} - b^{2}$	$2 a + b$

The sum of any two sides exceeds the third, so the third side satisfies $∣ a - b ∣ < x < a + b$ .
In similar triangles the ratio of areas is the square of the ratio of corresponding sides.
The triangle formed by joining the mid-points has half the sides (a quarter of the area).

Example 1 · Basic

Find the area of a triangle with sides 10 cm, 13 cm and 13 cm.

Solution, step by step

1 $s = \frac{10 + 13 + 13}{2} = 18$ .
2 $18 \times 8 \times 5 \times 5 = 3600 = 60$ sq cm

Answer 60 sq cm

Example 2 · Standard

An equilateral triangle has perimeter 60 cm. Find its area.

Solution, step by step

1Side $= 20$ cm.
2Area $= \frac{3}{4} \times 400 = 1003$ sq cm

Answer $1003$ sq cm

BasicTry it yourself first

The sides of a triangle are 7 cm, 15 cm and 20 cm. Find its area in sq cm.

StandardTry it yourself first

The sides of a triangle are 20 cm, 21 cm and 29 cm. Find its area in sq cm.

StandardTry it yourself first

The sides of a triangle are 9 cm, 10 cm and 17 cm. Find its area in sq cm.

StandardTry it yourself first

The sides of a triangle are 17 cm, 25 cm and 26 cm. Find its area in sq cm.

AdvancedTry it yourself first

The sides of a triangle are 14 cm, 25 cm and 25 cm. Find its area in sq cm.

AdvancedTry it yourself first

The sides of a triangle are 13 cm, 13 cm and 24 cm. Find its area in sq cm.

AdvancedTry it yourself first

The sides of a triangle are 16 cm, 17 cm and 17 cm. Find its area in sq cm.

AdvancedTry it yourself first

The sides of a triangle are 6 cm, 25 cm and 29 cm. Find its area in sq cm.

BasicTry it yourself first

The side of an equilateral triangle is 10 cm. Find its area.

StandardTry it yourself first

The side of an equilateral triangle is 12 cm. Find its area.

StandardTry it yourself first

The side of an equilateral triangle is 20 cm. Find its area.

StandardTry it yourself first

The side of an equilateral triangle is 10 cm. Find its altitude.

StandardTry it yourself first

The side of an equilateral triangle is 14 cm. Find its altitude.

AdvancedTry it yourself first

The perimeter of an equilateral triangle is 72 cm. Find its area.

AdvancedTry it yourself first

The perimeter of an equilateral triangle is 48 cm. Find its area.

StandardTry it yourself first

Two sides of a triangle are 8 and 13 units. Which statement about the third side

x

is correct?

StandardTry it yourself first

Two sides of a triangle are 12 and 20 units. Which statement about the third side

x

is correct?

AdvancedTry it yourself first

Two sides of a triangle are 9 and 14 units. Which statement about the third side

x

is correct?

AdvancedTry it yourself first

Two sides of a triangle are 7 and 18 units. Which statement about the third side

x

is correct?

StandardTry it yourself first

Corresponding sides of two similar triangles are in the ratio 2 : 5. Find the ratio of their areas.

StandardTry it yourself first

Corresponding sides of two similar triangles are in the ratio 4 : 9. Find the ratio of their areas.

AdvancedTry it yourself first

Corresponding sides of two similar triangles are in the ratio 5 : 8. Find the ratio of their areas.

StandardTry it yourself first

A triangle has perimeter 32 cm. A smaller triangle is made by joining the mid-points of its sides, and then another by joining the mid-points of that smaller triangle. Find the perimeter of the smallest triangle (cm).

AdvancedTry it yourself first

A triangle has perimeter 64 cm. A smaller triangle is made by joining the mid-points of its sides, and then another by joining the mid-points of that smaller triangle. Find the perimeter of the smallest triangle (cm).

AdvancedTry it yourself first

A triangle has perimeter 80 cm. A smaller triangle is made by joining the mid-points of its sides, and then another by joining the mid-points of that smaller triangle. Find the perimeter of the smallest triangle (cm).

Right-angled and isosceles

The Pythagoras rule

Hypotenuse, perpendicular, base and the altitude on the hypotenuse; the height of an isosceles triangle.

Pythagoras: $h^{2} = p^{2} + b^{2}$ . Famous triples: $(3, 4, 5), (5, 12, 13), (8, 15, 17), (7, 24, 25), (20, 21, 29)$ and their multiples.
Altitude on the hypotenuse: write the area two ways — $\frac{1}{2} p b = \frac{1}{2} h \times altitude$ ⟹ altitude $= \frac{p b}{h}$ .
Isosceles: the altitude bisects the base: $h = a^{2} - (\frac{b}{2})^{2}$ .
If the sum of the equal sides is given the third side $=$ perimeter $-$ that sum.

Example 3 · Basic

The sides containing the right angle of a right-angled triangle are 9 cm and 12 cm. Find the altitude on the hypotenuse.

Solution, step by step

1Hypotenuse $= 81 + 144 = 15$ ; area $= \frac{1}{2} \times 9 \times 12 = 54$ .
2 $\frac{1}{2} \times 15 \times altitude = 54$ ⟹ altitude $= 7.2$ cm

Answer 7.2 cm

Example 4 · Standard

An isosceles triangle has equal sides of 13 cm and a base of 10 cm. Find its area.

Solution, step by step

1 $h = 1 3^{2} - 5^{2} = 144 = 12$ .
2Area $= \frac{1}{2} \times 10 \times 12 = 60$ sq cm

Answer 60 sq cm

BasicTry it yourself first

The two sides of a right-angled triangle containing the right angle are 9 cm and 12 cm. Find the hypotenuse.

BasicTry it yourself first

The two sides of a right-angled triangle containing the right angle are 8 cm and 15 cm. Find the hypotenuse.

StandardTry it yourself first

The two sides of a right-angled triangle containing the right angle are 7 cm and 24 cm. Find the hypotenuse.

StandardTry it yourself first

The two sides of a right-angled triangle containing the right angle are 20 cm and 21 cm. Find the hypotenuse.

AdvancedTry it yourself first

The two sides of a right-angled triangle containing the right angle are 12 cm and 35 cm. Find the hypotenuse.

AdvancedTry it yourself first

The two sides of a right-angled triangle containing the right angle are 28 cm and 45 cm. Find the hypotenuse.

BasicTry it yourself first

The area of a right-angled triangle is 60 sq cm and its perpendicular is 8 cm. Find its base in cm.

BasicTry it yourself first

The area of a right-angled triangle is 48 sq cm and its perpendicular is 12 cm. Find its base in cm.

StandardTry it yourself first

The area of a right-angled triangle is 84 sq cm and its perpendicular is 14 cm. Find its base in cm.

StandardTry it yourself first

The area of a right-angled triangle is 90 sq cm and its perpendicular is 20 cm. Find its base in cm.

AdvancedTry it yourself first

The area of a right-angled triangle is 150 sq cm and its perpendicular is 25 cm. Find its base in cm.

AdvancedTry it yourself first

The area of a right-angled triangle is 120 sq cm and its perpendicular is 16 cm. Find its base in cm.

StandardTry it yourself first

The sides containing the right angle of a right-angled triangle are 3 cm and 4 cm. Find the length of the altitude on the hypotenuse (cm).

StandardTry it yourself first

The sides containing the right angle of a right-angled triangle are 6 cm and 8 cm. Find the length of the altitude on the hypotenuse (cm).

StandardTry it yourself first

The sides containing the right angle of a right-angled triangle are 15 cm and 20 cm. Find the length of the altitude on the hypotenuse (cm).

AdvancedTry it yourself first

The sides containing the right angle of a right-angled triangle are 12 cm and 16 cm. Find the length of the altitude on the hypotenuse (cm).

AdvancedTry it yourself first

The sides containing the right angle of a right-angled triangle are 7 cm and 24 cm. Find the length of the altitude on the hypotenuse (cm).

AdvancedTry it yourself first

The sides containing the right angle of a right-angled triangle are 24 cm and 32 cm. Find the length of the altitude on the hypotenuse (cm).

BasicTry it yourself first

An isosceles triangle has perimeter 36 cm and its two equal sides together measure 22 cm. Find the third side (cm).

BasicTry it yourself first

An isosceles triangle has perimeter 28 cm and its two equal sides together measure 16 cm. Find the third side (cm).

StandardTry it yourself first

An isosceles triangle has perimeter 50 cm and its two equal sides together measure 30 cm. Find the third side (cm).

StandardTry it yourself first

An isosceles triangle has perimeter 60 cm and its two equal sides together measure 36 cm. Find the third side (cm).

StandardTry it yourself first

An isosceles triangle has equal sides of 5 cm and a base of 6 cm. Find its area in sq cm.

StandardTry it yourself first

An isosceles triangle has equal sides of 10 cm and a base of 12 cm. Find its area in sq cm.

AdvancedTry it yourself first

An isosceles triangle has equal sides of 17 cm and a base of 16 cm. Find its area in sq cm.

AdvancedTry it yourself first

An isosceles triangle has equal sides of 25 cm and a base of 14 cm. Find its area in sq cm.

AdvancedTry it yourself first

An isosceles triangle has equal sides of 20 cm and a base of 24 cm. Find its area in sq cm.

AdvancedTry it yourself first

An isosceles triangle has equal sides of 15 cm and a base of 18 cm. Find its area in sq cm.

Quadrilaterals

From the rectangle to the trapezium

Area and perimeter of rectangles, squares, parallelograms, rhombuses and trapeziums.

Rectangles and squares

l b

and

a^{2}

Area, perimeter, diagonal and ratio-based questions.

◭ Diagram

Rectangle:

A = l \times b

P = 2 (l + b)

, diagonal

d = l^{2} + b^{2}

Figure	Area	Perimeter	Diagonal
Rectangle	$l b$	$2 (l + b)$	$l^{2} + b^{2}$
Square	$a^{2} = \frac{d ^{2}}{2}$	$4 a$	$a 2$

Given a ratio: length $p x$ , breadth $q x$ ; get $x$ from the difference or sum.
Given the half-perimeter and the difference, $l = \frac{S + D}{2}$ , $b = \frac{S - D}{2}$ .
If the square’s area doubles, the diagonal becomes $2$ times.

Example 5 · Basic

The length of a rectangle exceeds its breadth by 6 m and its perimeter is 52 m. Find its area.

Solution, step by step

1 $l + b = 26$ , $l - b = 6$ ⟹ $l = 16$ , $b = 10$ .
2Area $= 16 \times 10 = 160$ sq m

Answer 160 sq m

Example 6 · Standard

One side of a rectangle is 12 m and its diagonal is 20 m. Find its area.

Solution, step by step

1The other side $= 400 - 144 = 16$ .
2Area $= 12 \times 16 = 192$ sq m

Answer 192 sq m

BasicTry it yourself first

A rectangle is 6 cm long and 8 cm wide. Find its diagonal (cm).

BasicTry it yourself first

A rectangle is 5 cm long and 12 cm wide. Find its diagonal (cm).

StandardTry it yourself first

A rectangle is 9 cm long and 12 cm wide. Find its diagonal (cm).

StandardTry it yourself first

A rectangle is 8 cm long and 15 cm wide. Find its diagonal (cm).

AdvancedTry it yourself first

A rectangle is 7 cm long and 24 cm wide. Find its diagonal (cm).

AdvancedTry it yourself first

A rectangle is 20 cm long and 21 cm wide. Find its diagonal (cm).

StandardTry it yourself first

One side of a rectangle is 5 m and its diagonal is 13 m. Find its area in sq m.

StandardTry it yourself first

One side of a rectangle is 7 m and its diagonal is 25 m. Find its area in sq m.

StandardTry it yourself first

One side of a rectangle is 8 m and its diagonal is 17 m. Find its area in sq m.

AdvancedTry it yourself first

One side of a rectangle is 9 m and its diagonal is 15 m. Find its area in sq m.

AdvancedTry it yourself first

One side of a rectangle is 10 m and its diagonal is 26 m. Find its area in sq m.

AdvancedTry it yourself first

One side of a rectangle is 20 m and its diagonal is 29 m. Find its area in sq m.

StandardTry it yourself first

The length and breadth of a rectangle are in the ratio 7 : 4 and the length is 12 m more than the breadth. Find its area in sq m.

StandardTry it yourself first

The length and breadth of a rectangle are in the ratio 3 : 2 and the length is 6 m more than the breadth. Find its area in sq m.

AdvancedTry it yourself first

The length and breadth of a rectangle are in the ratio 5 : 2 and the length is 9 m more than the breadth. Find its area in sq m.

AdvancedTry it yourself first

The length and breadth of a rectangle are in the ratio 4 : 3 and the length is 5 m more than the breadth. Find its area in sq m.

AdvancedTry it yourself first

The length and breadth of a rectangle are in the ratio 9 : 5 and the length is 16 m more than the breadth. Find its area in sq m.

StandardTry it yourself first

The length of a rectangle exceeds its breadth by 6 m and its perimeter is 60 m. Find its area in sq m.

StandardTry it yourself first

The length of a rectangle exceeds its breadth by 11 m and its perimeter is 78 m. Find its area in sq m.

AdvancedTry it yourself first

The length of a rectangle exceeds its breadth by 14 m and its perimeter is 92 m. Find its area in sq m.

AdvancedTry it yourself first

The length of a rectangle exceeds its breadth by 2 m and its perimeter is 80 m. Find its area in sq m.

BasicTry it yourself first

A square has perimeter 52 cm. Find its area in sq cm.

BasicTry it yourself first

A square field has area 144 sq m. Find its perimeter (m).

StandardTry it yourself first

The diagonal of a square is

72

cm. Find its area in sq cm.

StandardTry it yourself first

The diagonal of a square is

52

cm. Find the diagonal of another square whose area is double that of the first. (cm)

StandardTry it yourself first

Find the area of a square inscribed in a circle of radius 6 cm (sq cm).

AdvancedTry it yourself first

Find the area of a square inscribed in a circle of radius 9 cm (sq cm).

AdvancedTry it yourself first

The diagonal of a square is

112

cm. Find its area in sq cm.

AdvancedTry it yourself first

The diagonal of a square is

82

cm. Find the diagonal of another square whose area is double that of the first. (cm)

Parallelograms, rhombuses and trapeziums

Height and diagonals

Base × height; product of diagonals / 2; mean of the parallel sides × height.

◭ Diagram

Parallelogram:

A = b \times h

(

h

the perpendicular distance)

◭ Diagram

Trapezium:

A = \frac{1}{2} (a + b) h

Parallelogram: $A = b \times h$ ; perimeter $2 (a + b)$ ; the diagonals bisect each other.
Rhombus: all four sides equal; $A = \frac{1}{2} d_{1} d_{2}$ ; the diagonals bisect each other at right angles, so side $= (\frac{d _{1}}{2})^{2} + (\frac{d _{2}}{2})^{2}$ .
Trapezium: if the sum and the difference of the parallel sides are known both can be found.

Example 7 · Basic

The diagonals of a rhombus are 16 cm and 30 cm. Find its perimeter.

Solution, step by step

1Side $= 8^{2} + 1 5^{2} = 17$ .
2Perimeter $= 4 \times 17 = 68$ cm

Answer 68 cm

Example 8 · Standard

A trapezium has area 270 sq cm, the difference of its parallel sides is 6 cm and the distance between them is 10 cm. Find the parallel sides.

Solution, step by step

1Sum $= \frac{2 \times 270}{10} = 54$ ; difference $= 6$ .
2Larger $= 30$ , smaller $= 24$

Answer 30 cm and 24 cm

BasicTry it yourself first

The base of a parallelogram is 4 times its height and its area is 144 sq cm. Find the height (cm).

StandardTry it yourself first

The base of a parallelogram is 3 times its height and its area is 432 sq cm. Find the height (cm).

StandardTry it yourself first

The base of a parallelogram is 5 times its height and its area is 320 sq cm. Find the height (cm).

AdvancedTry it yourself first

The base of a parallelogram is 2 times its height and its area is 450 sq cm. Find the height (cm).

BasicTry it yourself first

The diagonals of a rhombus are 6 cm and 8 cm. Find its perimeter (cm).

StandardTry it yourself first

The diagonals of a rhombus are 10 cm and 24 cm. Find its perimeter (cm).

StandardTry it yourself first

The diagonals of a rhombus are 14 cm and 48 cm. Find its perimeter (cm).

AdvancedTry it yourself first

The diagonals of a rhombus are 18 cm and 24 cm. Find its perimeter (cm).

AdvancedTry it yourself first

The diagonals of a rhombus are 30 cm and 40 cm. Find its perimeter (cm).

BasicTry it yourself first

The diagonals of a rhombus are 12 m and 9 m. Find its area in sq m.

StandardTry it yourself first

The diagonals of a rhombus are 16 m and 11 m. Find its area in sq m.

AdvancedTry it yourself first

The diagonals of a rhombus are 24 m and 18 m. Find its area in sq m.

BasicTry it yourself first

The parallel sides of a trapezium are 12 m and 8 m and the distance between them is 5 m. Find its area in sq m.

StandardTry it yourself first

The parallel sides of a trapezium are 18 m and 14 m and the distance between them is 9 m. Find its area in sq m.

AdvancedTry it yourself first

The parallel sides of a trapezium are 25 m and 15 m and the distance between them is 12 m. Find its area in sq m.

StandardTry it yourself first

A trapezium has area 300 sq m and parallel sides 24 m and 16 m. Find the distance between them (m).

StandardTry it yourself first

A trapezium has area 225 sq m and parallel sides 18 m and 12 m. Find the distance between them (m).

AdvancedTry it yourself first

A trapezium has area 400 sq m and parallel sides 25 m and 15 m. Find the distance between them (m).

StandardTry it yourself first

The difference of the parallel sides of a trapezium is 6 cm and the distance between them is 10 cm. If the area is 250 sq cm, find the parallel sides.

StandardTry it yourself first

The difference of the parallel sides of a trapezium is 10 cm and the distance between them is 12 cm. If the area is 540 sq cm, find the parallel sides.

AdvancedTry it yourself first

The difference of the parallel sides of a trapezium is 4 cm and the distance between them is 9 cm. If the area is 198 sq cm, find the parallel sides.

AdvancedTry it yourself first

The difference of the parallel sides of a trapezium is 8 cm and the distance between them is 15 cm. If the area is 540 sq cm, find the parallel sides.

Polygons

Angles and diagonals

The angles of a regular polygon, the number of diagonals and the area of a hexagon.

Angles, diagonals and the hexagon

Exterior angles add up to

36 0^{\circ}

All the formulas for a regular polygon of $n$ sides.

Quantity	Formula
Sum of interior angles	$(n - 2) \times 18 0^{\circ}$
Each interior angle (regular)	$\frac{( n - 2 ) 18 0 ^{\circ}}{n}$
Each exterior angle (regular)	$\frac{36 0 ^{\circ}}{n}$
Number of diagonals	$\frac{n ( n - 3 )}{2}$
Area of a regular hexagon	$\frac{3 3}{2} a^{2}$

An easy thing to remember: interior angle + exterior angle $= 18 0^{\circ}$ at every vertex. Walking round the polygon the exterior angles make one full turn ( $36 0^{\circ}$ ), which gives $\frac{360}{n}$ . Joining a regular hexagon to its centre gives 6 equilateral triangles.

Example 9 · Basic

Find each exterior and each interior angle of a regular octagon (8 sides).

Solution, step by step

1Exterior $= \frac{360}{8} = 4 5^{\circ}$ .
2Interior $= 180 - 45 = 13 5^{\circ}$

Answer 45° and 135°

Example 10 · Standard

Each exterior angle of a regular polygon is

2 4^{\circ}

. Find the number of sides and of diagonals.

Solution, step by step

1Sides $= \frac{360}{24} = 15$ .
2Diagonals $= \frac{15 \times 12}{2} = 90$

Answer 15 sides; 90 diagonals

BasicTry it yourself first

Find each interior angle of a regular polygon with 6 sides.

BasicTry it yourself first

Find each interior angle of a regular polygon with 5 sides.

StandardTry it yourself first

Find each interior angle of a regular polygon with 9 sides.

StandardTry it yourself first

Find each interior angle of a regular polygon with 10 sides.

AdvancedTry it yourself first

Find each interior angle of a regular polygon with 12 sides.

BasicTry it yourself first

Find each exterior angle of a regular polygon with 5 sides.

StandardTry it yourself first

Find each exterior angle of a regular polygon with 9 sides.

StandardTry it yourself first

Find each exterior angle of a regular polygon with 10 sides.

AdvancedTry it yourself first

Find each exterior angle of a regular polygon with 20 sides.

StandardTry it yourself first

Each exterior angle of a regular polygon is 18°. How many sides does it have?

StandardTry it yourself first

Each exterior angle of a regular polygon is 20°. How many sides does it have?

AdvancedTry it yourself first

Each exterior angle of a regular polygon is 15°. How many sides does it have?

AdvancedTry it yourself first

Each exterior angle of a regular polygon is 10°. How many sides does it have?

BasicTry it yourself first

How many diagonals does a convex polygon with 8 sides have?

StandardTry it yourself first

How many diagonals does a convex polygon with 10 sides have?

StandardTry it yourself first

How many diagonals does a convex polygon with 12 sides have?

AdvancedTry it yourself first

How many diagonals does a convex polygon with 14 sides have?

AdvancedTry it yourself first

How many diagonals does a convex polygon with 20 sides have?

StandardTry it yourself first

Find the area of a regular hexagon of side 10 cm (sq cm).

StandardTry it yourself first

Find the area of a regular hexagon of side 12 cm (sq cm).

AdvancedTry it yourself first

Find the area of a regular hexagon of side 8 cm (sq cm).

AdvancedTry it yourself first

Find the area of a regular hexagon of side 14 cm (sq cm).

AdvancedTry it yourself first

Find the area of a regular hexagon of side 20 cm (sq cm).

The circle

The game of

π

Circle, semicircle, sector, arc, ring, wheel and race track.

Circle, semicircle and sector

π = \frac{22}{7}

When the radius is a multiple of 7, $π = \frac{22}{7}$ is convenient.

◭ Diagram

Circle:

C = 2 π r

A = π r^{2}

; sector: area

\frac{θ}{360} π r^{2}

, arc

\frac{θ}{360} 2 π r

Semicircle: area $\frac{1}{2} π r^{2}$ ; perimeter $π r + 2 r$ (arc $+$ diameter) — count the straight edge too.
Sector: the fraction $\frac{θ}{360}$ of the whole circle (for both area and arc).
Given the circumference $r = \frac{C}{2 π}$ ; given the area $r = A / π$ .
A cow on a rope: the grazing area is a circle with the rope as radius.

Example 11 · Basic

Find the perimeter of a semicircle of radius 14 cm. (

π = \frac{22}{7}

)

Solution, step by step

1Arc $= π r = \frac{22}{7} \times 14 = 44$ ; diameter $= 28$ .
2Perimeter $= 44 + 28 = 72$ cm

Answer 72 cm

Example 12 · Standard

Find the area of a sector of radius 21 cm and central angle

12 0^{\circ}

. (

π = \frac{22}{7}

)

Solution, step by step

1Full circle $= \frac{22}{7} \times 441 = 1386$ .
2Sector $= \frac{120}{360} \times 1386 = 462$ sq cm

Answer 462 sq cm

BasicTry it yourself first

A circular field has radius 7 m. Find its area in sq m (

π = \frac{22}{7}

BasicTry it yourself first

A circular field has radius 14 m. Find its area in sq m (

π = \frac{22}{7}

StandardTry it yourself first

A circular field has radius 21 m. Find its area in sq m (

π = \frac{22}{7}

AdvancedTry it yourself first

A circular field has radius 28 m. Find its area in sq m (

π = \frac{22}{7}

BasicTry it yourself first

Find the circumference of a circle of radius 7 cm (

π = \frac{22}{7}

StandardTry it yourself first

Find the circumference of a circle of radius 14 cm (

π = \frac{22}{7}

StandardTry it yourself first

Find the circumference of a circle of radius 35 cm (

π = \frac{22}{7}

AdvancedTry it yourself first

Find the circumference of a circle of radius 21 cm (

π = \frac{22}{7}

StandardTry it yourself first

Find the perimeter of a semicircle of radius 7 cm (

π = \frac{22}{7}

StandardTry it yourself first

Find the perimeter of a semicircle of radius 21 cm (

π = \frac{22}{7}

AdvancedTry it yourself first

Find the perimeter of a semicircle of radius 28 cm (

π = \frac{22}{7}

StandardTry it yourself first

Find the area of a semicircle of radius 14 cm (

π = \frac{22}{7}

AdvancedTry it yourself first

Find the area of a semicircle of radius 21 cm (

π = \frac{22}{7}

StandardTry it yourself first

A circular garden has area 154 sq m. Find its radius (m,

π = \frac{22}{7}

StandardTry it yourself first

A circular garden has area 616 sq m. Find its radius (m,

π = \frac{22}{7}

AdvancedTry it yourself first

A circular garden has area 1386 sq m. Find its radius (m,

π = \frac{22}{7}

StandardTry it yourself first

A circle has circumference 44 cm. Find its radius (cm,

π = \frac{22}{7}

AdvancedTry it yourself first

A circle has circumference 176 cm. Find its radius (cm,

π = \frac{22}{7}

StandardTry it yourself first

Find the area of a sector with radius 7 cm and central angle 90° (sq cm,

π = \frac{22}{7}

StandardTry it yourself first

Find the area of a sector with radius 14 cm and central angle 45° (sq cm,

π = \frac{22}{7}

AdvancedTry it yourself first

Find the area of a sector with radius 21 cm and central angle 60° (sq cm,

π = \frac{22}{7}

AdvancedTry it yourself first

Find the area of a sector with radius 35 cm and central angle 36° (sq cm,

π = \frac{22}{7}

StandardTry it yourself first

Find the length of an arc with radius 14 cm and central angle 90° (cm,

π = \frac{22}{7}

StandardTry it yourself first

Find the length of an arc with radius 21 cm and central angle 60° (cm,

π = \frac{22}{7}

AdvancedTry it yourself first

Find the length of an arc with radius 28 cm and central angle 45° (cm,

π = \frac{22}{7}

AdvancedTry it yourself first

Find the length of an arc with radius 35 cm and central angle 72° (cm,

π = \frac{22}{7}

StandardTry it yourself first

A cow is tied to a peg with a rope so that it can graze 616 sq m. How long is the rope (m,

π = \frac{22}{7}

AdvancedTry it yourself first

A cow is tied to a peg with a rope so that it can graze 1386 sq m. How long is the rope (m,

π = \frac{22}{7}

Ring, wheel and track

The difference of two circles

The region between two concentric circles; wheel revolutions and track width.

◭ Diagram

Ring:

A = π (R^{2} - r^{2})

; difference of the circumferences

= 2 π (R - r)

Ring: area of the bigger circle $-$ area of the smaller.
Track of width $w$ : outer radius $=$ inner radius $+ w$ .
Wheel: the distance in one revolution $=$ circumference $= \frac{total distance}{revolutions}$ .

Example 13 · Standard

A circular track 7 m wide has an inner circumference of 220 m. Find the radius of the outer circle. (

π = \frac{22}{7}

)

Solution, step by step

1Inner radius $= \frac{220 \times 7}{2 \times 22} = 35$ m.
2Outer $= 35 + 7 = 42$ m

Answer 42 m

Example 14 · Standard

A wheel covers 11 km in 1000 revolutions. Find its radius (m). (

π = \frac{22}{7}

)

Solution, step by step

1Per revolution $= \frac{11000}{1000} = 11$ m (this is the circumference).
2 $2 π r = 11$ ⟹ $r = 1.75$ m

Answer 1.75 m

StandardTry it yourself first

Two concentric circles have radii 14 cm and 7 cm. Find the area of the ring between them (sq cm,

π = \frac{22}{7}

StandardTry it yourself first

Two concentric circles have radii 21 cm and 14 cm. Find the area of the ring between them (sq cm,

π = \frac{22}{7}

AdvancedTry it yourself first

Two concentric circles have radii 28 cm and 21 cm. Find the area of the ring between them (sq cm,

π = \frac{22}{7}

AdvancedTry it yourself first

Two concentric circles have radii 35 cm and 28 cm. Find the area of the ring between them (sq cm,

π = \frac{22}{7}

StandardTry it yourself first

A circular race track 5 m wide has an inner circumference of 132 m. Find the radius of the outer circle (m,

π = \frac{22}{7}

StandardTry it yourself first

A circular race track 3 m wide has an inner circumference of 176 m. Find the radius of the outer circle (m,

π = \frac{22}{7}

AdvancedTry it yourself first

A circular race track 4 m wide has an inner circumference of 264 m. Find the radius of the outer circle (m,

π = \frac{22}{7}

AdvancedTry it yourself first

A circular race track 6 m wide has an inner circumference of 88 m. Find the radius of the outer circle (m,

π = \frac{22}{7}

AdvancedTry it yourself first

A circular race track 7 m wide has an inner circumference of 308 m. Find the radius of the outer circle (m,

π = \frac{22}{7}

StandardTry it yourself first

A wheel makes 5,000 revolutions in covering 22 km. Find its radius (m,

π = \frac{22}{7}

StandardTry it yourself first

A wheel makes 500 revolutions in covering 11 km. Find its radius (m,

π = \frac{22}{7}

AdvancedTry it yourself first

A wheel makes 2,500 revolutions in covering 22 km. Find its radius (m,

π = \frac{22}{7}

AdvancedTry it yourself first

A wheel makes 400 revolutions in covering 5.28 km. Find its radius (m,

π = \frac{22}{7}

AdvancedTry it yourself first

A wheel makes 1,000 revolutions in covering 2.2 km. Find its radius (m,

π = \frac{22}{7}

Percent changes

Area when sides change

How much the area changes when the sides change by $a %$ and $b %$ .

$a + b + \frac{ab}{100}$

A decrease is negative

One formula for combining two changes, whether increase or decrease.

∑ Formula

क्षेत्रफल में परिवर्तन = a + b + \frac{ab}{100} %

Length changes by

a %

and breadth by

b %

(use a negative for a decrease)

Square / circle (one side or radius changes): change in area $= 2 a + \frac{a ^{2}}{100} %$ .
Perimeter, diagonal and circumference change by the same $a %$ as all the sides.
Think as multipliers: $+ 20%$ means $\times 1.20$ , $- 10%$ means $\times 0.90$ .

Example 15 · Basic

The length of a rectangle is increased by 20% and the breadth by 10%. By what percent does the area increase?

Solution, step by step

1 $\times 1.2 \times 1.1 = 1.32$ .
2Increase $= 32%$ (rule: $20 + 10 + \frac{200}{100} = 32$ ✓)

Answer 32%

Example 16 · Standard

The radius of a circle is increased by 20%. By what percent does the area increase?

Solution, step by step

1Area $\propto r^{2}$ : $1. 2^{2} = 1.44$ .
2Increase $= 44%$

Answer 44%

BasicTry it yourself first

The length of a rectangle is increased by 20% and its breadth by 15%. By what percent does its area increase?

StandardTry it yourself first

The length of a rectangle is increased by 25% and its breadth by 20%. By what percent does its area increase?

StandardTry it yourself first

The length of a rectangle is increased by 12% and its breadth by 10%. By what percent does its area increase?

AdvancedTry it yourself first

The length of a rectangle is increased by 30% and its breadth by 10%. By what percent does its area increase?

AdvancedTry it yourself first

The length of a rectangle is increased by 40% and its breadth by 25%. By what percent does its area increase?

StandardTry it yourself first

The length of a rectangle is increased by 20% and the breadth decreased by 10%. By what percent does the area increase?

StandardTry it yourself first

The length of a rectangle is increased by 30% and the breadth decreased by 20%. By what percent does the area increase?

AdvancedTry it yourself first

The length of a rectangle is increased by 25% and the breadth decreased by 10%. By what percent does the area increase?

AdvancedTry it yourself first

The length of a rectangle is increased by 40% and the breadth decreased by 25%. By what percent does the area increase?

BasicTry it yourself first

If all the sides of a figure are increased by 15%, by what percent does its perimeter increase?

StandardTry it yourself first

The side of a square is increased by 10%. By what percent does its area increase?

StandardTry it yourself first

The side of a square is increased by 20%. By what percent does its area increase?

AdvancedTry it yourself first

The side of a square is increased by 25%. By what percent does its area increase?

AdvancedTry it yourself first

The side of a square is increased by 12%. By what percent does its area increase?

StandardTry it yourself first

The radius of a circle is increased by 10%. By what percent does the area increase?

StandardTry it yourself first

The radius of a circle is increased by 5%. By what percent does the area increase?

AdvancedTry it yourself first

The radius of a circle is increased by 30%. By what percent does the area increase?

AdvancedTry it yourself first

The radius of a circle is increased by 25%. By what percent does the area increase?

AdvancedTry it yourself first

The radius of a circle is increased by 8%. By what percent does the area increase?

Paths, costs and inscribed figures

Everyday questions

A path round a field, roads, tiles, carpet, fencing and a figure drawn inside another.

Paths and costs

Outer

-

inner

The path’s area is the difference between the outer and inner figures; cost $=$ area $\times$ rate.

A path $x$ wide all round — outside: $2 x (l + b + 2 x)$ ; inside: $2 x (l + b - 2 x)$ .
Two roads through the middle (one along each side), each $x$ wide: $x (l + b - x)$ — the crossing square only once.
Carpet: carpet length $= \frac{room area}{carpet width}$ ; cost = length × rate.
Tiles: room area $\div$ area of one tile (same units). 1 hectare $= 10, 000$ sq m.

Example 17 · Standard

A path 3 m wide runs round the outside of a 60 m by 40 m rectangular field. Find the path’s area.

Solution, step by step

1Outer rectangle $66 \times 46 = 3036$ ; field $2400$ .
2Path $= 3036 - 2400 = 636$ sq m (formula $2 \times 3 \times 106 = 636$ ✓)

Answer 636 sq m

Example 18 · Standard

A path of equal width outside a 24 m by 14 m garden is to take up half the garden’s area. Find its width.

Solution, step by step

1Path $= \frac{336}{2} = 168$ ⟹ $2 x (38 + 2 x) = 168$ ⟹ $x^{2} + 19 x - 42 = 0$ .
2 $(x + 21) (x - 2) = 0$ ⟹ $x = 2$ m (the negative root is rejected)

Answer 2 m

BasicTry it yourself first

A path 2 m wide runs all around outside a 50 m by 30 m rectangular field. Find the area of the path (sq m).

StandardTry it yourself first

A path 4 m wide runs all around inside a 80 m by 50 m rectangular field. Find the area of the path (sq m).

StandardTry it yourself first

A path 5 m wide runs all around outside a 120 m by 90 m rectangular field. Find the area of the path (sq m).

AdvancedTry it yourself first

A path 2 m wide runs all around inside a 45 m by 30 m rectangular field. Find the area of the path (sq m).

AdvancedTry it yourself first

A path 6 m wide runs all around inside a 100 m by 70 m rectangular field. Find the area of the path (sq m).

StandardTry it yourself first

A 90 m by 70 m rectangular lawn has two roads, each 5 m wide, through its middle — one parallel to the length and one to the breadth. Find the cost of laying bricks on the roads at ₹12 per sq m (in ₹).

StandardTry it yourself first

A 120 m by 80 m rectangular lawn has two roads, each 6 m wide, through its middle — one parallel to the length and one to the breadth. Find the cost of laying bricks on the roads at ₹10 per sq m (in ₹).

AdvancedTry it yourself first

A 100 m by 60 m rectangular lawn has two roads, each 4 m wide, through its middle — one parallel to the length and one to the breadth. Find the cost of laying bricks on the roads at ₹20 per sq m (in ₹).

AdvancedTry it yourself first

A 75 m by 50 m rectangular lawn has two roads, each 3 m wide, through its middle — one parallel to the length and one to the breadth. Find the cost of laying bricks on the roads at ₹15 per sq m (in ₹).

StandardTry it yourself first

A path of equal width is to be made outside a 28 m by 24 m garden so that it takes up

\frac{1}{3}

of the garden’s area. Find the width of the path (m).

StandardTry it yourself first

A path of equal width is to be made outside a 36 m by 32 m garden so that it takes up

\frac{1}{4}

of the garden’s area. Find the width of the path (m).

AdvancedTry it yourself first

A path of equal width is to be made outside a 44 m by 40 m garden so that it takes up

\frac{1}{5}

of the garden’s area. Find the width of the path (m).

AdvancedTry it yourself first

A path of equal width is to be made outside a 30 m by 24 m garden so that it takes up

\frac{1}{2}

of the garden’s area. Find the width of the path (m).

AdvancedTry it yourself first

A path of equal width is to be made outside a 42 m by 36 m garden so that it takes up

\frac{1}{3}

of the garden’s area. Find the width of the path (m).

AdvancedTry it yourself first

A path of equal width is to be made outside a 54 m by 48 m garden so that it takes up

\frac{1}{4}

of the garden’s area. Find the width of the path (m).

BasicTry it yourself first

The floor of a square room of side 12 m is to be covered completely with square tiles of side 40 cm. What is the smallest number of tiles needed?

StandardTry it yourself first

The floor of a square room of side 8 m is to be covered completely with square tiles of side 25 cm. What is the smallest number of tiles needed?

StandardTry it yourself first

The floor of a square room of side 15 m is to be covered completely with square tiles of side 75 cm. What is the smallest number of tiles needed?

AdvancedTry it yourself first

The floor of a square room of side 20 m is to be covered completely with square tiles of side 40 cm. What is the smallest number of tiles needed?

AdvancedTry it yourself first

The floor of a square room of side 12 m is to be covered completely with square tiles of side 50 cm. What is the smallest number of tiles needed?

StandardTry it yourself first

Find the total cost of carpeting a room 9 m long and 5 m broad with a carpet 75 cm wide at ₹25 per metre (in ₹).

StandardTry it yourself first

Find the total cost of carpeting a room 12 m long and 8 m broad with a carpet 80 cm wide at ₹30 per metre (in ₹).

AdvancedTry it yourself first

Find the total cost of carpeting a room 10 m long and 6 m broad with a carpet 60 cm wide at ₹18 per metre (in ₹).

AdvancedTry it yourself first

Find the total cost of carpeting a room 15 m long and 8 m broad with a carpet 120 cm wide at ₹45 per metre (in ₹).

StandardTry it yourself first

Cultivating a square field at ₹4500 per hectare cost ₹18,000. What would it cost to fence it at ₹12 per metre (in ₹)?

AdvancedTry it yourself first

Cultivating a square field at ₹3000 per hectare cost ₹48,000. What would it cost to fence it at ₹15 per metre (in ₹)?

AdvancedTry it yourself first

Cultivating a square field at ₹2400 per hectare cost ₹60,000. What would it cost to fence it at ₹20 per metre (in ₹)?

AdvancedTry it yourself first

Cultivating a square field at ₹1800 per hectare cost ₹64,800. What would it cost to fence it at ₹25 per metre (in ₹)?

One figure inside another

Equal perimeter, largest inscribed figure

The largest circle in a rectangle, a square in a circle, a triangle in a semicircle; corner quadrants.

Largest circle in a rectangle: diameter $=$ the shorter side.
A square inscribed in a circle: the square’s diagonal $=$ the circle’s diameter; area $= 2 r^{2}$ .
Largest triangle in a semicircle: base $2 r$ , height $r$ , area $r^{2}$ .
Equal perimeters: the square’s perimeter = the circle’s circumference; that gives the circle’s radius.
Four corner quadrants make up one full circle.

Example 19 · Standard

A square has area 121 sq cm. Find the area of the circle formed by bending a wire of the same perimeter. (

π = \frac{22}{7}

)

Solution, step by step

1Side $= 11$ ; perimeter $= 44$ = circumference ⟹ $r = 7$ .
2Area $= \frac{22}{7} \times 49 = 154$ sq cm

Answer 154 sq cm

Example 20 · Advanced

A square park of side 60 m has a quadrant-shaped bed of radius 7 m at each corner. Find the remaining area. (

π = \frac{22}{7}

)

Solution, step by step

1Four quadrants $=$ one full circle $= \frac{22}{7} \times 49 = 154$ .
2Remaining $= 3600 - 154 = 3446$ sq m

Answer 3446 sq m

StandardTry it yourself first

A square has area 484 sq cm. Find the area of the circle formed by bending a wire of the same perimeter (sq cm,

π = \frac{22}{7}

StandardTry it yourself first

A square has area 1089 sq cm. Find the area of the circle formed by bending a wire of the same perimeter (sq cm,

π = \frac{22}{7}

AdvancedTry it yourself first

A square has area 1936 sq cm. Find the area of the circle formed by bending a wire of the same perimeter (sq cm,

π = \frac{22}{7}

AdvancedTry it yourself first

A square has area 3025 sq cm. Find the area of the circle formed by bending a wire of the same perimeter (sq cm,

π = \frac{22}{7}

StandardTry it yourself first

Find the area of the largest circle that can be drawn inside a 24 cm by 14 cm rectangle (sq cm,

π = \frac{22}{7}

StandardTry it yourself first

Find the area of the largest circle that can be drawn inside a 30 cm by 28 cm rectangle (sq cm,

π = \frac{22}{7}

AdvancedTry it yourself first

Find the area of the largest circle that can be drawn inside a 28 cm by 21 cm rectangle (sq cm,

π = \frac{22}{7}

AdvancedTry it yourself first

Find the area of the largest circle that can be drawn inside a 36 cm by 35 cm rectangle (sq cm,

π = \frac{22}{7}

StandardTry it yourself first

The largest triangle is inscribed in a semicircle of radius 7 cm. Find the area inside the semicircle not occupied by the triangle (sq cm,

π = \frac{22}{7}

StandardTry it yourself first

The largest triangle is inscribed in a semicircle of radius 14 cm. Find the area inside the semicircle not occupied by the triangle (sq cm,

π = \frac{22}{7}

AdvancedTry it yourself first

The largest triangle is inscribed in a semicircle of radius 21 cm. Find the area inside the semicircle not occupied by the triangle (sq cm,

π = \frac{22}{7}

StandardTry it yourself first

A square park of side 50 m has a quadrant-shaped flower bed of radius 7 m at each corner. Find the area of the remaining part (sq m,

π = \frac{22}{7}

AdvancedTry it yourself first

A square park of side 40 m has a quadrant-shaped flower bed of radius 14 m at each corner. Find the area of the remaining part (sq m,

π = \frac{22}{7}

AdvancedTry it yourself first

A square park of side 84 m has a quadrant-shaped flower bed of radius 14 m at each corner. Find the area of the remaining part (sq m,

π = \frac{22}{7}

AdvancedTry it yourself first

A square park of side 56 m has a quadrant-shaped flower bed of radius 21 m at each corner. Find the area of the remaining part (sq m,

π = \frac{22}{7}

Traps and fast tricks

Slips · tricks

Where marks go in area and perimeter questions.

The commonest mistakes

Seven traps

The slips that cost marks.

Trap 1Mixing the units of perimeter and area. Perimeter is in cm or m, area in sq cm or sq m. If tiles are in cm and the room in m, make the units alike first.

Trap 2Forgetting the diameter in a semicircle’s perimeter. It is

π r + 2 r

, not just

π r

Trap 3Using the slanting side as the height of a trapezium. The height is the perpendicular distance between the parallel sides.

Trap 4Taking a diagonal of a rhombus as its side. Use Pythagoras on the half-diagonals to get the side.

Trap 5Swapping the inside and outside formulas for a path. Outside is

+ 2 x

, inside

- 2 x

— a path all round takes the width twice on each dimension.

Trap 6Adding percent changes directly. If length rises 20% and breadth 10%, the area rises 32%, not 30%.

Trap 7Use $π = \frac{22}{7}$ when the radius is a multiple of 7. Otherwise use the value the question gives.

Fast tricks

Recognise and apply

Habits for speed.

Learn the Pythagorean triples: $3 - 4 - 5, 5 - 12 - 13, 8 - 15 - 17, 7 - 24 - 25, 20 - 21 - 29$ — they save the square roots.
For the same perimeter: rectangle $<$ square $<$ circle in area; the circle encloses the most.
For a square: $A = \frac{d ^{2}}{2}$ and $P = 4 a$ ; diagonal $= a 2$ .
If the sides become $k$ times, the perimeter becomes $k$ times but the area $k^{2}$ times.
Any regular polygon = $n$ equal triangles meeting at the centre.

Example 21 · Standard

A square and a circle have equal perimeters. The square’s side is 22 cm. By how much does the circle’s area exceed the square’s? (

π = \frac{22}{7}

)

Solution, step by step

1Perimeter $= 88$ ⟹ $2 π r = 88$ ⟹ $r = 14$ .
2Circle $= \frac{22}{7} \times 196 = 616$ ; square $= 484$ ; difference $= 132$ sq cm

Answer 132 sq cm

The practice ladder

Basic → standard → advanced

New questions of every type, in three levels.

Level 1 — Basic

Direct formulas

One-formula questions.

BasicTry it yourself first

The sides of a triangle are 5 cm, 12 cm and 13 cm. Find its area in sq cm.

BasicTry it yourself first

The sides of a triangle are 9 cm, 10 cm and 17 cm. Find its area in sq cm.

BasicTry it yourself first

The two sides of a right-angled triangle containing the right angle are 5 cm and 12 cm. Find the hypotenuse.

BasicTry it yourself first

The two sides of a right-angled triangle containing the right angle are 15 cm and 20 cm. Find the hypotenuse.

BasicTry it yourself first

The area of a right-angled triangle is 54 sq cm and its perpendicular is 9 cm. Find its base in cm.

BasicTry it yourself first

The area of a right-angled triangle is 96 sq cm and its perpendicular is 12 cm. Find its base in cm.

BasicTry it yourself first

The sides containing the right angle of a right-angled triangle are 8 cm and 6 cm. Find the length of the altitude on the hypotenuse (cm).

BasicTry it yourself first

An isosceles triangle has perimeter 44 cm and its two equal sides together measure 26 cm. Find the third side (cm).

BasicTry it yourself first

An isosceles triangle has equal sides of 13 cm and a base of 24 cm. Find its area in sq cm.

BasicTry it yourself first

The side of an equilateral triangle is 8 cm. Find its area.

BasicTry it yourself first

The side of an equilateral triangle is 6 cm. Find its altitude.

BasicTry it yourself first

A rectangle is 15 cm long and 20 cm wide. Find its diagonal (cm).

BasicTry it yourself first

One side of a rectangle is 6 m and its diagonal is 10 m. Find its area in sq m.

BasicTry it yourself first

The length and breadth of a rectangle are in the ratio 5 : 3 and the length is 8 m more than the breadth. Find its area in sq m.

BasicTry it yourself first

The length of a rectangle exceeds its breadth by 6 m and its perimeter is 44 m. Find its area in sq m.

BasicTry it yourself first

A square has perimeter 36 cm. Find its area in sq cm.

BasicTry it yourself first

A square field has area 225 sq m. Find its perimeter (m).

BasicTry it yourself first

The diagonal of a square is

62

cm. Find its area in sq cm.

BasicTry it yourself first

The base of a parallelogram is 3 times its height and its area is 75 sq cm. Find the height (cm).

BasicTry it yourself first

The diagonals of a rhombus are 24 cm and 32 cm. Find its perimeter (cm).

BasicTry it yourself first

The diagonals of a rhombus are 14 m and 10 m. Find its area in sq m.

BasicTry it yourself first

The parallel sides of a trapezium are 14 m and 10 m and the distance between them is 6 m. Find its area in sq m.

BasicTry it yourself first

A trapezium has area 136 sq m and parallel sides 20 m and 14 m. Find the distance between them (m).

BasicTry it yourself first

Find each exterior angle of a regular polygon with 12 sides.

BasicTry it yourself first

Each exterior angle of a regular polygon is 40°. How many sides does it have?

BasicTry it yourself first

How many diagonals does a convex polygon with 6 sides have?

BasicTry it yourself first

Find the area of a regular hexagon of side 6 cm (sq cm).

BasicTry it yourself first

A circular field has radius 35 m. Find its area in sq m (

π = \frac{22}{7}

BasicTry it yourself first

Find the circumference of a circle of radius 28 cm (

π = \frac{22}{7}

BasicTry it yourself first

Find the area of a semicircle of radius 7 cm (

π = \frac{22}{7}

BasicTry it yourself first

Find the perimeter of a semicircle of radius 14 cm (

π = \frac{22}{7}

BasicTry it yourself first

A circle has circumference 88 cm. Find its radius (cm,

π = \frac{22}{7}

BasicTry it yourself first

A circular garden has area 1386 sq m. Find its radius (m,

π = \frac{22}{7}

BasicTry it yourself first

Find the area of a sector with radius 14 cm and central angle 90° (sq cm,

π = \frac{22}{7}

BasicTry it yourself first

Find the length of an arc with radius 21 cm and central angle 120° (cm,

π = \frac{22}{7}

BasicTry it yourself first

Two concentric circles have radii 10 cm and 3 cm. Find the area of the ring between them (sq cm,

π = \frac{22}{7}

BasicTry it yourself first

A cow is tied to a peg with a rope so that it can graze 2464 sq m. How long is the rope (m,

π = \frac{22}{7}

BasicTry it yourself first

A circular race track 7 m wide has an inner circumference of 176 m. Find the radius of the outer circle (m,

π = \frac{22}{7}

BasicTry it yourself first

A wheel makes 100 revolutions in covering 0.44 km. Find its radius (m,

π = \frac{22}{7}

BasicTry it yourself first

The length of a rectangle is increased by 10% and its breadth by 10%. By what percent does its area increase?

BasicTry it yourself first

The side of a square is increased by 30%. By what percent does its area increase?

BasicTry it yourself first

The radius of a circle is increased by 50%. By what percent does the area increase?

BasicTry it yourself first

If all the sides of a figure are increased by 8%, by what percent does its perimeter increase?

BasicTry it yourself first

A path 2 m wide runs all around outside a 40 m by 25 m rectangular field. Find the area of the path (sq m).

BasicTry it yourself first

A path 1 m wide runs all around inside a 36 m by 28 m rectangular field. Find the area of the path (sq m).

BasicTry it yourself first

The floor of a square room of side 6 m is to be covered completely with square tiles of side 30 cm. What is the smallest number of tiles needed?

BasicTry it yourself first

Find the total cost of carpeting a room 8 m long and 5 m broad with a carpet 50 cm wide at ₹20 per metre (in ₹).

BasicTry it yourself first

Cultivating a square field at ₹6000 per hectare cost ₹6,000. What would it cost to fence it at ₹10 per metre (in ₹)?

BasicTry it yourself first

Two sides of a triangle are 5 and 9 units. Which statement about the third side

x

is correct?

BasicTry it yourself first

Corresponding sides of two similar triangles are in the ratio 1 : 3. Find the ratio of their areas.

BasicTry it yourself first

A square has area 5929 sq cm. Find the area of the circle formed by bending a wire of the same perimeter (sq cm,

π = \frac{22}{7}

Level 2 — Standard

Two ideas combined

Exam-level questions.

StandardTry it yourself first

The sides of a triangle are 10 cm, 17 cm and 21 cm. Find its area in sq cm.

StandardTry it yourself first

The sides of a triangle are 12 cm, 35 cm and 37 cm. Find its area in sq cm.

StandardTry it yourself first

The sides of a triangle are 15 cm, 15 cm and 24 cm. Find its area in sq cm.

StandardTry it yourself first

The two sides of a right-angled triangle containing the right angle are 16 cm and 30 cm. Find the hypotenuse.

StandardTry it yourself first

The area of a right-angled triangle is 210 sq cm and its perpendicular is 20 cm. Find its base in cm.

StandardTry it yourself first

The area of a right-angled triangle is 168 sq cm and its perpendicular is 16 cm. Find its base in cm.

StandardTry it yourself first

The sides containing the right angle of a right-angled triangle are 18 cm and 24 cm. Find the length of the altitude on the hypotenuse (cm).

StandardTry it yourself first

An isosceles triangle has perimeter 72 cm and its two equal sides together measure 40 cm. Find the third side (cm).

StandardTry it yourself first

An isosceles triangle has equal sides of 29 cm and a base of 40 cm. Find its area in sq cm.

StandardTry it yourself first

An isosceles triangle has equal sides of 17 cm and a base of 30 cm. Find its area in sq cm.

StandardTry it yourself first

The side of an equilateral triangle is 18 cm. Find its area.

StandardTry it yourself first

The side of an equilateral triangle is 30 cm. Find its altitude.

StandardTry it yourself first

The perimeter of an equilateral triangle is 54 cm. Find its area.

StandardTry it yourself first

A rectangle is 12 cm long and 35 cm wide. Find its diagonal (cm).

StandardTry it yourself first

One side of a rectangle is 9 m and its diagonal is 41 m. Find its area in sq m.

StandardTry it yourself first

One side of a rectangle is 12 m and its diagonal is 37 m. Find its area in sq m.

StandardTry it yourself first

The length and breadth of a rectangle are in the ratio 8 : 5 and the length is 18 m more than the breadth. Find its area in sq m.

StandardTry it yourself first

The length of a rectangle exceeds its breadth by 14 m and its perimeter is 100 m. Find its area in sq m.

StandardTry it yourself first

The diagonal of a square is

122

cm. Find its area in sq cm.

StandardTry it yourself first

The diagonal of a square is

92

cm. Find the diagonal of another square whose area is double that of the first. (cm)

StandardTry it yourself first

Find the area of a square inscribed in a circle of radius 10 cm (sq cm).

StandardTry it yourself first

The base of a parallelogram is 4 times its height and its area is 196 sq cm. Find the height (cm).

StandardTry it yourself first

The diagonals of a rhombus are 20 cm and 48 cm. Find its perimeter (cm).

StandardTry it yourself first

The diagonals of a rhombus are 36 cm and 48 cm. Find its perimeter (cm).

StandardTry it yourself first

The diagonals of a rhombus are 28 m and 24 m. Find its area in sq m.

StandardTry it yourself first

The parallel sides of a trapezium are 32 m and 22 m and the distance between them is 15 m. Find its area in sq m.

StandardTry it yourself first

A trapezium has area 375 sq m and parallel sides 28 m and 22 m. Find the distance between them (m).

StandardTry it yourself first

The difference of the parallel sides of a trapezium is 8 cm and the distance between them is 14 cm. If the area is 448 sq cm, find the parallel sides.

StandardTry it yourself first

The difference of the parallel sides of a trapezium is 5 cm and the distance between them is 8 cm. If the area is 196 sq cm, find the parallel sides.

StandardTry it yourself first

Find each interior angle of a regular polygon with 20 sides.

StandardTry it yourself first

Find each exterior angle of a regular polygon with 16 sides.

StandardTry it yourself first

Each exterior angle of a regular polygon is 12°. How many sides does it have?

StandardTry it yourself first

Each exterior angle of a regular polygon is 8°. How many sides does it have?

StandardTry it yourself first

How many diagonals does a convex polygon with 16 sides have?

StandardTry it yourself first

Find the area of a regular hexagon of side 18 cm (sq cm).

StandardTry it yourself first

A circular field has radius 42 m. Find its area in sq m (

π = \frac{22}{7}

StandardTry it yourself first

Find the circumference of a circle of radius 49 cm (

π = \frac{22}{7}

StandardTry it yourself first

Find the perimeter of a semicircle of radius 35 cm (

π = \frac{22}{7}

StandardTry it yourself first

Find the area of a semicircle of radius 28 cm (

π = \frac{22}{7}

StandardTry it yourself first

A circular garden has area 2464 sq m. Find its radius (m,

π = \frac{22}{7}

StandardTry it yourself first

A circle has circumference 132 cm. Find its radius (cm,

π = \frac{22}{7}

StandardTry it yourself first

Find the area of a sector with radius 28 cm and central angle 135° (sq cm,

π = \frac{22}{7}

StandardTry it yourself first

Find the area of a sector with radius 21 cm and central angle 150° (sq cm,

π = \frac{22}{7}

StandardTry it yourself first

Find the length of an arc with radius 42 cm and central angle 30° (cm,

π = \frac{22}{7}

StandardTry it yourself first

Two concentric circles have radii 42 cm and 35 cm. Find the area of the ring between them (sq cm,

π = \frac{22}{7}

StandardTry it yourself first

A cow is tied to a peg with a rope so that it can graze 3850 sq m. How long is the rope (m,

π = \frac{22}{7}

StandardTry it yourself first

A circular race track 8 m wide has an inner circumference of 440 m. Find the radius of the outer circle (m,

π = \frac{22}{7}

StandardTry it yourself first

A circular race track 5 m wide has an inner circumference of 154 m. Find the radius of the outer circle (m,

π = \frac{22}{7}

StandardTry it yourself first

A wheel makes 2,000 revolutions in covering 8.8 km. Find its radius (m,

π = \frac{22}{7}

StandardTry it yourself first

A wheel makes 700 revolutions in covering 15.4 km. Find its radius (m,

π = \frac{22}{7}

StandardTry it yourself first

The length of a rectangle is increased by 15% and its breadth by 20%. By what percent does its area increase?

StandardTry it yourself first

The length of a rectangle is increased by 35% and its breadth by 20%. By what percent does its area increase?

StandardTry it yourself first

The length of a rectangle is increased by 50% and the breadth decreased by 20%. By what percent does the area increase?

StandardTry it yourself first

The length of a rectangle is increased by 60% and the breadth decreased by 30%. By what percent does the area increase?

StandardTry it yourself first

The side of a square is increased by 40%. By what percent does its area increase?

StandardTry it yourself first

The side of a square is increased by 15%. By what percent does its area increase?

StandardTry it yourself first

The radius of a circle is increased by 40%. By what percent does the area increase?

StandardTry it yourself first

The radius of a circle is increased by 15%. By what percent does the area increase?

StandardTry it yourself first

A path 3 m wide runs all around outside a 70 m by 45 m rectangular field. Find the area of the path (sq m).

StandardTry it yourself first

A path 4 m wide runs all around inside a 90 m by 60 m rectangular field. Find the area of the path (sq m).

StandardTry it yourself first

A path 5 m wide runs all around outside a 55 m by 35 m rectangular field. Find the area of the path (sq m).

StandardTry it yourself first

A 80 m by 60 m rectangular lawn has two roads, each 4 m wide, through its middle — one parallel to the length and one to the breadth. Find the cost of laying bricks on the roads at ₹15 per sq m (in ₹).

StandardTry it yourself first

A 110 m by 90 m rectangular lawn has two roads, each 5 m wide, through its middle — one parallel to the length and one to the breadth. Find the cost of laying bricks on the roads at ₹18 per sq m (in ₹).

StandardTry it yourself first

A path of equal width is to be made outside a 14 m by 12 m garden so that it takes up

\frac{1}{3}

of the garden’s area. Find the width of the path (m).

StandardTry it yourself first

A path of equal width is to be made outside a 22 m by 20 m garden so that it takes up

\frac{1}{5}

of the garden’s area. Find the width of the path (m).

StandardTry it yourself first

A path of equal width is to be made outside a 32 m by 12 m garden so that it takes up

\frac{1}{2}

of the garden’s area. Find the width of the path (m).

StandardTry it yourself first

The floor of a square room of side 9 m is to be covered completely with square tiles of side 30 cm. What is the smallest number of tiles needed?

StandardTry it yourself first

The floor of a square room of side 10 m is to be covered completely with square tiles of side 50 cm. What is the smallest number of tiles needed?

StandardTry it yourself first

The floor of a square room of side 18 m is to be covered completely with square tiles of side 60 cm. What is the smallest number of tiles needed?

StandardTry it yourself first

Find the total cost of carpeting a room 14 m long and 6 m broad with a carpet 70 cm wide at ₹24 per metre (in ₹).

StandardTry it yourself first

Find the total cost of carpeting a room 16 m long and 9 m broad with a carpet 90 cm wide at ₹35 per metre (in ₹).

StandardTry it yourself first

Cultivating a square field at ₹3600 per hectare cost ₹32,400. What would it cost to fence it at ₹18 per metre (in ₹)?

StandardTry it yourself first

Cultivating a square field at ₹4800 per hectare cost ₹30,000. What would it cost to fence it at ₹16 per metre (in ₹)?

StandardTry it yourself first

A square has area 4356 sq cm. Find the area of the circle formed by bending a wire of the same perimeter (sq cm,

π = \frac{22}{7}

StandardTry it yourself first

Find the area of the largest circle that can be drawn inside a 42 cm by 28 cm rectangle (sq cm,

π = \frac{22}{7}

StandardTry it yourself first

Find the area of the largest circle that can be drawn inside a 25 cm by 14 cm rectangle (sq cm,

π = \frac{22}{7}

StandardTry it yourself first

The largest triangle is inscribed in a semicircle of radius 28 cm. Find the area inside the semicircle not occupied by the triangle (sq cm,

π = \frac{22}{7}

StandardTry it yourself first

A square park of side 70 m has a quadrant-shaped flower bed of radius 7 m at each corner. Find the area of the remaining part (sq m,

π = \frac{22}{7}

StandardTry it yourself first

A square park of side 100 m has a quadrant-shaped flower bed of radius 14 m at each corner. Find the area of the remaining part (sq m,

π = \frac{22}{7}

StandardTry it yourself first

Two sides of a triangle are 6 and 15 units. Which statement about the third side

x

is correct?

StandardTry it yourself first

Two sides of a triangle are 11 and 16 units. Which statement about the third side

x

is correct?

StandardTry it yourself first

Corresponding sides of two similar triangles are in the ratio 3 : 7. Find the ratio of their areas.

StandardTry it yourself first

A triangle has perimeter 48 cm. A smaller triangle is made by joining the mid-points of its sides, and then another by joining the mid-points of that smaller triangle. Find the perimeter of the smallest triangle (cm).

Level 3 — Advanced

Many steps and clever thinking

Questions at the hard end of competitive exams.

AdvancedTry it yourself first

The sides of a triangle are 25 cm, 39 cm and 40 cm. Find its area in sq cm.

AdvancedTry it yourself first

The sides of a triangle are 18 cm, 24 cm and 30 cm. Find its area in sq cm.

AdvancedTry it yourself first

The sides of a triangle are 29 cm, 29 cm and 40 cm. Find its area in sq cm.

AdvancedTry it yourself first

The two sides of a right-angled triangle containing the right angle are 33 cm and 56 cm. Find the hypotenuse.

AdvancedTry it yourself first

The two sides of a right-angled triangle containing the right angle are 36 cm and 77 cm. Find the hypotenuse.

AdvancedTry it yourself first

The area of a right-angled triangle is 330 sq cm and its perpendicular is 22 cm. Find its base in cm.

AdvancedTry it yourself first

The area of a right-angled triangle is 270 sq cm and its perpendicular is 18 cm. Find its base in cm.

AdvancedTry it yourself first

The sides containing the right angle of a right-angled triangle are 21 cm and 28 cm. Find the length of the altitude on the hypotenuse (cm).

AdvancedTry it yourself first

The sides containing the right angle of a right-angled triangle are 27 cm and 36 cm. Find the length of the altitude on the hypotenuse (cm).

AdvancedTry it yourself first

An isosceles triangle has perimeter 98 cm and its two equal sides together measure 62 cm. Find the third side (cm).

AdvancedTry it yourself first

An isosceles triangle has equal sides of 37 cm and a base of 24 cm. Find its area in sq cm.

AdvancedTry it yourself first

An isosceles triangle has equal sides of 41 cm and a base of 18 cm. Find its area in sq cm.

AdvancedTry it yourself first

An isosceles triangle has equal sides of 26 cm and a base of 20 cm. Find its area in sq cm.

AdvancedTry it yourself first

The side of an equilateral triangle is 22 cm. Find its area.

AdvancedTry it yourself first

The side of an equilateral triangle is 26 cm. Find its altitude.

AdvancedTry it yourself first

The perimeter of an equilateral triangle is 102 cm. Find its area.

AdvancedTry it yourself first

Two sides of a triangle are 13 and 21 units. Which statement about the third side

x

is correct?

AdvancedTry it yourself first

Two sides of a triangle are 8 and 19 units. Which statement about the third side

x

is correct?

AdvancedTry it yourself first

A rectangle is 28 cm long and 45 cm wide. Find its diagonal (cm).

AdvancedTry it yourself first

A rectangle is 33 cm long and 56 cm wide. Find its diagonal (cm).

AdvancedTry it yourself first

One side of a rectangle is 11 m and its diagonal is 61 m. Find its area in sq m.

AdvancedTry it yourself first

One side of a rectangle is 16 m and its diagonal is 65 m. Find its area in sq m.

AdvancedTry it yourself first

The length and breadth of a rectangle are in the ratio 11 : 7 and the length is 12 m more than the breadth. Find its area in sq m.

AdvancedTry it yourself first

The length and breadth of a rectangle are in the ratio 13 : 8 and the length is 15 m more than the breadth. Find its area in sq m.

AdvancedTry it yourself first

The length of a rectangle exceeds its breadth by 18 m and its perimeter is 144 m. Find its area in sq m.

AdvancedTry it yourself first

The length of a rectangle exceeds its breadth by 17 m and its perimeter is 174 m. Find its area in sq m.

AdvancedTry it yourself first

A square has perimeter 100 cm. Find its area in sq cm.

AdvancedTry it yourself first

A square field has area 1024 sq m. Find its perimeter (m).

AdvancedTry it yourself first

Find the area of a square inscribed in a circle of radius 13 cm (sq cm).

AdvancedTry it yourself first

The diagonal of a square is

142

cm. Find the diagonal of another square whose area is double that of the first. (cm)

AdvancedTry it yourself first

The base of a parallelogram is 5 times its height and its area is 405 sq cm. Find the height (cm).

AdvancedTry it yourself first

The base of a parallelogram is 6 times its height and its area is 726 sq cm. Find the height (cm).

AdvancedTry it yourself first

The diagonals of a rhombus are 48 cm and 14 cm. Find its perimeter (cm).

AdvancedTry it yourself first

The diagonals of a rhombus are 60 cm and 80 cm. Find its perimeter (cm).

AdvancedTry it yourself first

The diagonals of a rhombus are 42 m and 36 m. Find its area in sq m.

AdvancedTry it yourself first

The parallel sides of a trapezium are 38 m and 26 m and the distance between them is 17 m. Find its area in sq m.

AdvancedTry it yourself first

A trapezium has area 640 sq m and parallel sides 42 m and 38 m. Find the distance between them (m).

AdvancedTry it yourself first

A trapezium has area 540 sq m and parallel sides 35 m and 25 m. Find the distance between them (m).

AdvancedTry it yourself first

The difference of the parallel sides of a trapezium is 9 cm and the distance between them is 16 cm. If the area is 648 sq cm, find the parallel sides.

AdvancedTry it yourself first

The difference of the parallel sides of a trapezium is 7 cm and the distance between them is 12 cm. If the area is 354 sq cm, find the parallel sides.

AdvancedTry it yourself first

Find each interior angle of a regular polygon with 18 sides.

AdvancedTry it yourself first

Find each interior angle of a regular polygon with 15 sides.

AdvancedTry it yourself first

Find each exterior angle of a regular polygon with 40 sides.

AdvancedTry it yourself first

Each exterior angle of a regular polygon is 6°. How many sides does it have?

AdvancedTry it yourself first

Each exterior angle of a regular polygon is 5°. How many sides does it have?

AdvancedTry it yourself first

How many diagonals does a convex polygon with 25 sides have?

AdvancedTry it yourself first

How many diagonals does a convex polygon with 30 sides have?

AdvancedTry it yourself first

Find the area of a regular hexagon of side 24 cm (sq cm).

AdvancedTry it yourself first

Find the area of a regular hexagon of side 30 cm (sq cm).

AdvancedTry it yourself first

A circular field has radius 49 m. Find its area in sq m (

π = \frac{22}{7}

AdvancedTry it yourself first

Find the circumference of a circle of radius 63 cm (

π = \frac{22}{7}

AdvancedTry it yourself first

Find the perimeter of a semicircle of radius 42 cm (

π = \frac{22}{7}

AdvancedTry it yourself first

Find the area of a semicircle of radius 35 cm (

π = \frac{22}{7}

AdvancedTry it yourself first

A circular garden has area 3850 sq m. Find its radius (m,

π = \frac{22}{7}

AdvancedTry it yourself first

A circle has circumference 264 cm. Find its radius (cm,

π = \frac{22}{7}

AdvancedTry it yourself first

Find the area of a sector with radius 42 cm and central angle 210° (sq cm,

π = \frac{22}{7}

AdvancedTry it yourself first

Find the area of a sector with radius 14 cm and central angle 135° (sq cm,

π = \frac{22}{7}

AdvancedTry it yourself first

Find the length of an arc with radius 63 cm and central angle 40° (cm,

π = \frac{22}{7}

AdvancedTry it yourself first

Find the length of an arc with radius 49 cm and central angle 45° (cm,

π = \frac{22}{7}

AdvancedTry it yourself first

Two concentric circles have radii 49 cm and 42 cm. Find the area of the ring between them (sq cm,

π = \frac{22}{7}

AdvancedTry it yourself first

Two concentric circles have radii 63 cm and 56 cm. Find the area of the ring between them (sq cm,

π = \frac{22}{7}

AdvancedTry it yourself first

A cow is tied to a peg with a rope so that it can graze 7546 sq m. How long is the rope (m,

π = \frac{22}{7}

AdvancedTry it yourself first

A circular race track 9 m wide has an inner circumference of 308 m. Find the radius of the outer circle (m,

π = \frac{22}{7}

AdvancedTry it yourself first

A circular race track 10 m wide has an inner circumference of 352 m. Find the radius of the outer circle (m,

π = \frac{22}{7}

AdvancedTry it yourself first

A wheel makes 1,250 revolutions in covering 11 km. Find its radius (m,

π = \frac{22}{7}

AdvancedTry it yourself first

A wheel makes 3,000 revolutions in covering 26.4 km. Find its radius (m,

π = \frac{22}{7}

AdvancedTry it yourself first

The length of a rectangle is increased by 45% and its breadth by 40%. By what percent does its area increase?

AdvancedTry it yourself first

The length of a rectangle is increased by 18% and its breadth by 15%. By what percent does its area increase?

AdvancedTry it yourself first

The length of a rectangle is increased by 75% and the breadth decreased by 20%. By what percent does the area increase?

AdvancedTry it yourself first

The length of a rectangle is increased by 80% and the breadth decreased by 30%. By what percent does the area increase?

AdvancedTry it yourself first

The side of a square is increased by 35%. By what percent does its area increase?

AdvancedTry it yourself first

The side of a square is increased by 18%. By what percent does its area increase?

AdvancedTry it yourself first

The radius of a circle is increased by 45%. By what percent does the area increase?

AdvancedTry it yourself first

The radius of a circle is increased by 12%. By what percent does the area increase?

AdvancedTry it yourself first

A path 7 m wide runs all around outside a 150 m by 100 m rectangular field. Find the area of the path (sq m).

AdvancedTry it yourself first

A path 8 m wide runs all around inside a 130 m by 90 m rectangular field. Find the area of the path (sq m).

AdvancedTry it yourself first

A path 6 m wide runs all around outside a 85 m by 65 m rectangular field. Find the area of the path (sq m).

AdvancedTry it yourself first

A 140 m by 100 m rectangular lawn has two roads, each 7 m wide, through its middle — one parallel to the length and one to the breadth. Find the cost of laying bricks on the roads at ₹22 per sq m (in ₹).

AdvancedTry it yourself first

A 160 m by 120 m rectangular lawn has two roads, each 8 m wide, through its middle — one parallel to the length and one to the breadth. Find the cost of laying bricks on the roads at ₹25 per sq m (in ₹).

AdvancedTry it yourself first

A path of equal width is to be made outside a 36 m by 20 m garden so that it takes up

\frac{1}{3}

of the garden’s area. Find the width of the path (m).

AdvancedTry it yourself first

A path of equal width is to be made outside a 50 m by 40 m garden so that it takes up

\frac{1}{2}

of the garden’s area. Find the width of the path (m).

AdvancedTry it yourself first

A path of equal width is to be made outside a 66 m by 60 m garden so that it takes up

\frac{1}{5}

of the garden’s area. Find the width of the path (m).

AdvancedTry it yourself first

The floor of a square room of side 24 m is to be covered completely with square tiles of side 60 cm. What is the smallest number of tiles needed?

AdvancedTry it yourself first

The floor of a square room of side 21 m is to be covered completely with square tiles of side 35 cm. What is the smallest number of tiles needed?

AdvancedTry it yourself first

The floor of a square room of side 30 m is to be covered completely with square tiles of side 75 cm. What is the smallest number of tiles needed?

AdvancedTry it yourself first

Find the total cost of carpeting a room 18 m long and 12 m broad with a carpet 150 cm wide at ₹55 per metre (in ₹).

AdvancedTry it yourself first

Find the total cost of carpeting a room 20 m long and 14 m broad with a carpet 80 cm wide at ₹42 per metre (in ₹).

AdvancedTry it yourself first

Cultivating a square field at ₹2700 per hectare cost ₹1,72,800. What would it cost to fence it at ₹30 per metre (in ₹)?

AdvancedTry it yourself first

Cultivating a square field at ₹3600 per hectare cost ₹1,76,400. What would it cost to fence it at ₹24 per metre (in ₹)?

AdvancedTry it yourself first

A square has area 7744 sq cm. Find the area of the circle formed by bending a wire of the same perimeter (sq cm,

π = \frac{22}{7}

AdvancedTry it yourself first

A square has area 9801 sq cm. Find the area of the circle formed by bending a wire of the same perimeter (sq cm,

π = \frac{22}{7}

AdvancedTry it yourself first

Find the area of the largest circle that can be drawn inside a 49 cm by 56 cm rectangle (sq cm,

π = \frac{22}{7}

AdvancedTry it yourself first

Find the area of the largest circle that can be drawn inside a 63 cm by 42 cm rectangle (sq cm,

π = \frac{22}{7}

AdvancedTry it yourself first

The largest triangle is inscribed in a semicircle of radius 35 cm. Find the area inside the semicircle not occupied by the triangle (sq cm,

π = \frac{22}{7}

AdvancedTry it yourself first

The largest triangle is inscribed in a semicircle of radius 42 cm. Find the area inside the semicircle not occupied by the triangle (sq cm,

π = \frac{22}{7}

AdvancedTry it yourself first

A square park of side 126 m has a quadrant-shaped flower bed of radius 21 m at each corner. Find the area of the remaining part (sq m,

π = \frac{22}{7}

AdvancedTry it yourself first

A square park of side 140 m has a quadrant-shaped flower bed of radius 28 m at each corner. Find the area of the remaining part (sq m,

π = \frac{22}{7}

AdvancedTry it yourself first

Corresponding sides of two similar triangles are in the ratio 7 : 12. Find the ratio of their areas.

AdvancedTry it yourself first

Corresponding sides of two similar triangles are in the ratio 9 : 14. Find the ratio of their areas.

AdvancedTry it yourself first

A triangle has perimeter 112 cm. A smaller triangle is made by joining the mid-points of its sides, and then another by joining the mid-points of that smaller triangle. Find the perimeter of the smallest triangle (cm).

AdvancedTry it yourself first

A triangle has perimeter 144 cm. A smaller triangle is made by joining the mid-points of its sides, and then another by joining the mid-points of that smaller triangle. Find the perimeter of the smallest triangle (cm).

Revision — formula cards

Flip each card and test yourself

Read the front, think, then flip.