Pariksha SaathiHeight and Distance Questions with Solutions
Almost every question in this topic reduces to one right triangle and one of three standard angles — 30°, 45°, or 60° — so the real skill is drawing the triangle correctly and picking the right trig ratio, not memorizing many formulas. Here’s the concept, four fully worked examples, and a few to try yourself.
The concept, quickly
- The three standard angles and their tangent values (the ratio used most often here): tan 30° = 1/√3, tan 45° = 1, tan 60° = √3.
- Angle of elevation is measured UP from the horizontal (looking up at something); angle of depression is measured DOWN from the horizontal (looking down at something) — both are measured from the horizontal line, not from the ground or the object.
- tan(angle) = opposite side / adjacent side, where "opposite" is the height (vertical) and "adjacent" is the horizontal distance from the base — draw the triangle first, label height and distance, then apply tan.
- For a slanted line like a ladder or a wire (the hypotenuse), use sin or cos instead of tan: cos(angle) = adjacent/hypotenuse when the angle is with the ground, sin(angle) = opposite/hypotenuse.
Worked examples
1. A pole casts a shadow of length 10√3 m when the sun’s angle of elevation is 30°. Find the height of the pole.
Solution
- tan(30°) = height / shadow length, so height = shadow length × tan(30°).
- tan(30°) = 1/√3, so height = 10√3 × (1/√3) = 10 m.
Answer: 10 m
2. The angle of elevation of the top of a tower from a point 50 m away from its base is 45°. Find the height of the tower.
Solution
- tan(45°) = height / distance, so height = distance × tan(45°).
- tan(45°) = 1, so height = 50 × 1 = 50 m.
Answer: 50 m
3. A ladder leans against a wall, making an angle of 60° with the ground. If the foot of the ladder is 5 m from the wall, find the length of the ladder.
Solution
- The ladder is the hypotenuse, and 5 m is the side adjacent to the 60° angle: cos(60°) = adjacent / hypotenuse.
- cos(60°) = 1/2, so hypotenuse = 5 / (1/2) = 10 m.
Answer: 10 m
4. From the top of a cliff 100 m high, the angle of depression of a boat is 30°. Find the distance of the boat from the base of the cliff.
Solution
- The angle of depression from the top equals the angle of elevation from the boat (alternate angles), so tan(30°) = height / distance.
- tan(30°) = 1/√3, so 1/√3 = 100 / distance, giving distance = 100√3 ≈ 173.2 m.
Answer: 100√3 m ≈ 173.2 m
Try these yourself
Draw the triangle, identify the angle and sides, then check against the answer.
1. A tower casts a shadow of length 20√3 m when the sun’s angle of elevation is 30°. Find the height of the tower.
Answer: 20 m
2. The angle of elevation of the top of a building from a point 40 m away from its base is 45°. Find the height of the building.
Answer: 40 m
3. A ladder leans against a wall, making an angle of 60° with the ground. If the foot of the ladder is 6 m from the wall, find the length of the ladder.
Answer: 12 m
Where this comes up
Height and Distance is tested directly in the Quantitative/Numerical Aptitude section of every exam Pariksha Saathi covers — SSC CGL, SSC MTS, SSC CHSL, IBPS PO, IBPS Clerk, SBI PO and SBI Clerk — most often in SSC CGL.
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