JEE Mains Maths · Height & Distance

Heights and Distances

Finding heights and distances from angles of elevation and depression, using right triangles and the tangent of each angle.

All Height & Distance notes JEE Mains Maths strategy

Why this matters

Fifteen PYQs, all multiple choice, from 2021 to 2023. Six sight a tower or a moving object from two points; five have one part of a pole or tower subtending an angle at a point; four spread the observers across a horizontal plane, north and west of a tower or around a park. Three ideas cover the page.

Concept 1 of 3: Two angles of elevation

Every sighting is a right triangle: height over horizontal distance is the tangent of the angle. With two sightings of the same height, write one equation for each and eliminate the unknown distance. For a moving object, the distance covered divided by the time gives its speed.

Definition

Angle of elevation $\theta$ , height $h$ , distance $d$ : $h=d\tan\theta$ .
The angle of depression from the top equals the angle of elevation from below.
$\tan15^\circ=2-\sqrt3$ , $\tan75^\circ=2+\sqrt3$ , $\tan\frac\pi8=\sqrt2-1$ .

Height from a sighting

h=d\tan\theta

Worked example

From two points 40 m apart in line with a tower, the angles of elevation of its top are

30^\circ

and

60^\circ

. Find its height.

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

JEE Mains · 2022 · 28 June 2022 · Q79Moderate

Example 1 · Height & Distance · Heights and Distances

Let

AB

and

PQ

be two vertical poles,

160\text{ }m

apart from each other. Let

C

be the middle point of

B

and Q, which are feet of these two poles. Let

\frac{\pi}{8}

and

\theta

be the angles of elevation from

C

P

and

A

, respectively. If the height of pole

PQ

is twice the height of pole

AB

, then

\tan^{2}\theta

is equal to

Depression is measured from the horizontal

The angle of depression is between the horizontal line at the observer's eye and the line of sight, not between the line of sight and the vertical wall.

Drill 5 more on two angles of elevation

Concept 2 of 3: The angle subtended by part of a tower

When part of a pole subtends an angle at a point, the angle is a difference of two elevations: the elevation of the top minus the elevation of the mark. Take the tangent of that difference with the subtraction formula, or add the known angles, as in

60^\circ+15^\circ=75^\circ

Definition

Angle subtended by the part between heights $h_1<h_2$ : $\beta-\alpha$ , with $\tan\alpha=\frac{h_1}d$ , $\tan\beta=\frac{h_2}d$ .
$\tan(\beta-\alpha)=\frac{\tan\beta-\tan\alpha}{1+\tan\beta\tan\alpha}$ .
Equal angles subtended by two parts: the double-angle formula.

Difference of elevations

\tan(\beta-\alpha)=\frac{\tan\beta-\tan\alpha}{1+\tan\beta\tan\alpha}

Worked example

A 10 m flagpole stands on a 10 m building. From a point 10 m from the base, what angle does the flagpole subtend?

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

JEE Mains · 2022 · 29 June 2022 · Q168Moderate

Example 2 · Height & Distance · Heights and Distances

From the base of a pole of height 20 meter, the angle of elevation of the top of a tower is

60^{\circ}

. The pole subtends an angle

30^{\circ}

at the top of the tower. Then the height of the tower is:

Subtended is not elevation

The angle a part subtends is the difference of two elevations, not the elevation of its top. Add or subtract the angles before taking a tangent.

Drill 4 more on the angle subtended by part of a tower

Concept 3 of 3: Observers spread over the ground

When observers stand north, south or west of a tower, their distances from the foot are sides of a right triangle on the ground. Find each distance from the height and its angle, then use Pythagoras in the horizontal plane. Equal angles of elevation from several points mean equal distances from the foot, so the foot is a circumcentre.

Definition

Each observer: distance from the foot $=h\cot\theta$ .
Perpendicular directions (north and west): the distance between observers is $\sqrt{d_1^2+d_2^2}$ .
Equal elevations from $A,B,C$ : the foot is the circumcentre, so $h=R\tan\theta$ .

Pythagoras on the ground

d_{12}=\sqrt{(h\cot\theta_1)^2+(h\cot\theta_2)^2}

Worked example

A 12 m tower is seen at

45^\circ

from a point due east and at

60^\circ

from a point due north. How far apart are the two points?

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

JEE Mains · 2022 · 29 July 2022 · Q71Moderate

Example 3 · Height & Distance · Heights and Distances

The angle of elevation of the top of a tower from a point

A

due north of it is

\alpha

and from a point

B

at a distance of 9 units due west of

A

\cos^{- 1}\left( \frac{3}{\sqrt{13}} \right)

. If the distance of the point

B

from the tower is 15 units, then

\cot\alpha

is equal to:

Draw the ground view

The observers' positions form a horizontal triangle separate from the vertical ones. Sketch the view from above before combining distances.

Drill 3 more on observers spread over the ground

Summary — formulas & gotchas at a glance

A revision cheat-sheet for the formulas and gotchas above. Click any concept name to jump back to its full explanation.

Formulas (3)

Two angles of elevation
Height from a sighting
$h=d\tan\theta$
The angle subtended by part of a tower
Difference of elevations
$\tan(\beta-\alpha)=\frac{\tan\beta-\tan\alpha}{1+\tan\beta\tan\alpha}$
Observers spread over the ground
Pythagoras on the ground
$d_{12}=\sqrt{(h\cot\theta_1)^2+(h\cot\theta_2)^2}$

Watch out for (3)

Depression is measured from the horizontal
→ Two angles of elevation
Subtended is not elevation
→ The angle subtended by part of a tower
Draw the ground view
→ Observers spread over the ground

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