Height and Distance

You need not climb a tower to find its height. If you stand some distance away and measure the angle at which you see its top, a right triangle forms — and $tan θ$ from the last chapter gives the height. That is “height and distance”. Almost every question has just three ingredients: a correct figure, a right triangle and ** $tan 3 0^{\circ}, tan 4 5^{\circ}, tan 6 0^{\circ}$ **.

tan 3 0^{\circ} = \frac{1}{3}, tan 4 5^{\circ} = 1, tan 6 0^{\circ} = 3; h = d tan θ

The most important values and formula

1

The basics

Line of sight, elevation and depression

The terms, how to draw the figure and the $tan$ values of the three angles.

Terms, the figure and the values

The first step of every question: a figure

The definitions of elevation and depression and the rules for the figure.

Angle of elevation

θ

: the angle between the line of sight to the top of the object and the horizontal;

tan θ = \frac{h}{d}

Angle of depression: when the object is below the horizontal; it equals the alternate angle of elevation from below

Angle	$3 0^{\circ}$	$4 5^{\circ}$	$6 0^{\circ}$
$tan$	$\frac{1}{3}$	$1$	$3$
$cot$	$3$	$1$	$\frac{1}{3}$
$sin$	$\frac{1}{2}$	$\frac{1}{2}$	$\frac{3}{2}$
$cos$	$\frac{3}{2}$	$\frac{1}{2}$	$\frac{1}{2}$

Draw it: a vertical line (the tower), a horizontal line (the ground) and the line of sight make a right triangle; mark the angle where the observer stands.
The angles of elevation and depression are always measured from the horizontal, never from the vertical, and both are acute.
Unless stated otherwise, the observer’s height is ignored.
With two angles, write the shared height $h$ in both right triangles; the distances are $h cot α$ and $h cot β$ .

Example 1 · Basic

From a point 18 m from the foot of a tower the angle of elevation of its top is

6 0^{\circ}

. Find the height of the tower.

Solution, step by step

1 $tan 6 0^{\circ} = \frac{h}{18}$ .
2 $h = 183$ m

Answer $183$ m

2

One-angle problems

One right triangle, one formula

Questions on height, distance, angle, shadows, ladders, broken trees and poles.

Height, distance and shadows

tan θ = \frac{h}{d}

Finding the third of height, distance and angle from the other two; shadow questions.

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Ladders, broken trees and poles

Hypotenuse questions:

sin

and

cos

When a length is the hypotenuse (ladder, wire, broken part) use $sin, cos$ .

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3

Two-angle problems

A shared height, two equations

Angles that change on moving closer or away, ships and lighthouses, building and tower, a flagstaff and an aeroplane.

Moving and two observations

h (cot α - cot β) = m

The observer moves towards the tower, the shadow grows or shrinks, or the depression angles of ships are given.

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Building and tower, flagstaff and aeroplane

Two objects, several angles

Looking at one object from the top of another; the length of something mounted on top; the speed of a flying aeroplane.

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4

Traps and fast tricks

Slips · tricks

Where marks go in height-and-distance 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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5

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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6

Revision — formula cards

Flip each card and test yourself

Read the front, think, then flip.

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The basics

Terms, the figure and the values

One-angle problems

Height, distance and shadows

Ladders, broken trees and poles

Two-angle problems

Moving and two observations

Building and tower, flagstaff and aeroplane

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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