Area and Perimeter

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.

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

1

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.

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.

2

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.

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Parallelograms, rhombuses and trapeziums

Height and diagonals

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

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3

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.

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4

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.

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Ring, wheel and track

The difference of two circles

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

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5

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.

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6

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.

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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.

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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.

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Fast tricks

Recognise and apply

Habits for speed.

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8

The practice ladder

Basic → standard → advanced

New questions of every type, in three levels.

Level 1 — Basic

Direct formulas

One-formula questions.

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Level 2 — Standard

Two ideas combined

Exam-level questions.

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Level 3 — Advanced

Many steps and clever thinking

Questions at the hard end of competitive exams.

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9

Revision — formula cards

Flip each card and test yourself

Read the front, think, then flip.

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Triangles

Types and formulas

Right-angled and isosceles

Quadrilaterals

Rectangles and squares

Parallelograms, rhombuses and trapeziums

Polygons

Angles, diagonals and the hexagon

The circle

Circle, semicircle and sector

Ring, wheel and track

Percent changes

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

Paths, costs and inscribed figures

Paths and costs

One figure inside another

Traps and fast tricks

The commonest mistakes

Fast tricks

The practice ladder

Level 1 — Basic

Level 2 — Standard

Level 3 — Advanced

Revision — formula cards

Free All Access Passevery mind map · Hindi + English · 12 months

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