SSC CGL Maths · Heights and Distances

One Line of Sight

One right triangle: an object's height, a distance along the ground and the angle between them, or a ladder, rope or wire that is the slant side.

All Heights and Distances notes

Why this matters

Ten questions in the bank, all EASY. Four use tan to find a height, a distance, a shadow or the sun's angle, and six give a ladder, a pole, a rope, a wire, a string or a ramp as the slant side. Each is one ratio at 30°, 45° or 60°, so the work is picking the right ratio, not the arithmetic.

Concept 1 of 2: Height and ground distance from one angle

A tower, its shadow or the ground up to an observer, and the line of sight make a right triangle. The height is the side opposite the angle and the ground distance the side next to it, so tan links them and nothing else is needed.

Definition

The angle of elevation is the angle between the level ground (the horizontal) and your line of sight when you look up at something.
The angle of depression is the angle between the horizontal and your line of sight when you look down. It is measured from the horizontal, never from the vertical.
The angle of depression from the top of a tower to a point on the ground equals the angle of elevation of the tower top from that point, because the two horizontals are parallel.
Height $=$ distance $\times \tan\theta$ , and distance $=$ height $\div \tan\theta$ . A shadow is the ground distance, with $\theta$ the sun's elevation.
The three values you need: $\tan 30^\circ = \frac{1}{\sqrt{3}}$ , $\tan 45^\circ = 1$ , $\tan 60^\circ = \sqrt{3}$ .
Fast route: at 45° the height equals the distance. At 60° the height is $\sqrt{3}$ times the distance, and at 30° the distance is $\sqrt{3}$ times the height. When the options are decimals, use $\sqrt{3} \approx 1.732$ .

Height and distance from the angle of elevation

h = d \tan\theta \qquad d = \frac{h}{\tan\theta} \qquad \tan\theta = \frac{h}{d}

hthe height of the object
dthe distance along the ground to its foot, or the length of its shadow
θthe angle of elevation (or of depression, seen from the top)

Worked example

From a point 45 m from the foot of a tower, the angle of elevation of its top is 30°. Find the height of the tower.

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

SSC CGL · 2025 · Tier I, 17 Sep 2025, Shift 1 · Q59Easy

Example 1 · Heights and Distances · One Line of Sight

A flagpole has a shadow measuring 10 meters. Given that the height of the flagpole is

10\sqrt{3}

meters, What is the angle at which the sun is elevated?

Measuring the angle of depression from the vertical

The angle of depression is between the horizontal and the line of sight. A depression of 30° means the line of sight makes 60° with the vertical, and using 60° gives the wrong triangle.

Swapping the 30° and 60° ratios

\tan 60^\circ = \sqrt{3}

and

\tan 30^\circ = \frac{1}{\sqrt{3}}

. A higher sun means a larger angle and a shorter shadow, so at 60° the shadow is shorter than the height.

Expecting an exact answer when the options are rounded

SSC sometimes asks for an approximate distance and prints

500\sqrt{3}

as 866. Work exactly, then multiply by 1.732 at the end.

Drill 3 more on height and ground distance from one angle

Concept 2 of 2: A ladder, rope or wire as the slant side

When the question gives the length of something slanting (a ladder, a rope, a kite string, a ramp, a wire), that length is the hypotenuse. Sin then links it to the height, and cos links it to the ground distance.

Definition

Height reached $=$ length $\times \sin\theta$ , where $\theta$ is the angle with the ground.
Ground distance $=$ length $\times \cos\theta$ : how far the foot is from the wall or the pole.
Going the other way, length $=$ height $\div \sin\theta$ or ground distance $\div \cos\theta$ .
The values you need: $\sin 30^\circ = \cos 60^\circ = \frac{1}{2}$ , $\sin 60^\circ = \cos 30^\circ = \frac{\sqrt{3}}{2}$ , $\sin 45^\circ = \cos 45^\circ = \frac{1}{\sqrt{2}}$ .
Fast route, the special triangles: in a 30°, 60°, 90° triangle the sides are in the ratio $1 : \sqrt{3} : 2$ , so the side opposite 30° is half the slant side. In a 45°, 45°, 90° triangle they are $1 : 1 : \sqrt{2}$ , so the slant side is $\sqrt{2}$ times the height.
An angle given with the wall is the complement of the angle with the ground: 30° with the wall is 60° with the ground.

Slant side, height and ground distance

h = L\sin\theta \qquad d = L\cos\theta \qquad L = \frac{h}{\sin\theta} = \frac{d}{\cos\theta}

Lthe slanting length: ladder, rope, wire or ramp
θthe angle the slant side makes with the ground

Worked example

A ladder 14 m long leans against a wall and makes an angle of 60° with the ground. How high up the wall does it reach, and how far is its foot from the wall?

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

SSC CGL · 2025 · Tier I, 23 Sep 2025, Shift 2 · Q55Easy

Example 2 · Heights and Distances · One Line of Sight

A kite is soaring at an altitude of 75 meters above the ground, with its string temporarily secured to a point on the ground. The angle of elevation of the string relative to the ground is 30 degrees. Given that the string remains taut, what is the length of the string?

Using tan when the slant length is given

When the length of the ladder or rope is known, it is the hypotenuse, so the ratio is sin (for the height) or cos (for the ground distance). Tan links height and ground distance only.

Reading the angle with the wall as the angle with the ground

A ladder at 30° to the wall is at 60° to the ground. Its height up the wall is then

L\sin 60^\circ

, not

L\sin 30^\circ

Rounding too early when the answer is to the nearest metre

5\sqrt{2}

is about 7.07, so it rounds to 7. Rounding

\sqrt{2}

to 1.5 first gives 7.5, which rounds to 8 and picks the wrong option.

Drill 5 more on a ladder, rope or wire as the slant side

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 (2)

Height and ground distance from one angle
Height and distance from the angle of elevation
$h = d \tan\theta \qquad d = \frac{h}{\tan\theta} \qquad \tan\theta = \frac{h}{d}$
A ladder, rope or wire as the slant side
Slant side, height and ground distance
$h = L\sin\theta \qquad d = L\cos\theta \qquad L = \frac{h}{\sin\theta} = \frac{d}{\cos\theta}$

Watch out for (6)

Measuring the angle of depression from the vertical
→ Height and ground distance from one angle
Swapping the 30° and 60° ratios
→ Height and ground distance from one angle
Expecting an exact answer when the options are rounded
→ Height and ground distance from one angle
Using tan when the slant length is given
→ A ladder, rope or wire as the slant side
Reading the angle with the wall as the angle with the ground
→ A ladder, rope or wire as the slant side
Rounding too early when the answer is to the nearest metre
→ A ladder, rope or wire as the slant side

Test yourself on Heights and Distances

10 past SSC CGL questions from this chapter, timed at 6 minutes and marked the way the exam marks it. You see your score and every answer the moment you finish. Free to start.

Take the Heights and Distances chapter test Drill all 10 questions

Related notes

Trigonometry: standard values →