Trigonometry · Topic 7 of 7

🏗️ Height & Distance — Advanced

7 exam question types, fully solved

When the observer looks down at something below the horizontal — a boat, a car, the foot of a building — the angle the line of sight makes with the horizontal is the angle of depression. Because the horizontal at the top and the ground are parallel, this depression angle is exactly equal to the angle of elevation measured back from the far point (alternate angles). That one fact turns every “looking down” problem into an ordinary elevation triangle.

Advanced problems simply stack two right triangles that share the same vertical height: two observers, a pole on a tower, or one object seen above and below eye level. You almost never need to find the height and the distance separately.

Master method — one height, two triangles

1. Mark the common height and write for each triangle.

2. Turn each into a base: .

3. Subtract the bases (observers on the same side) or add them (opposite sides) to eliminate the unknown.

For two observers a distance apart on the same side, two distinct angles give the clean result . On opposite sides that minus becomes a plus.

Remember: a depression of from the top equals an elevation of from the far point, so you can reuse all the standard ratios.

🗼Heights & Distances LabTower · observer · angle of elevation

What do you know?

Distance between observers d

Find h from the gap d

cot 30° = √3,   cot 60° = 1/√3

cot α − cot β = √3 − 1/√3 = (3 − 1)/√3 = 2/√3 ≈ 1.15

h = d / (cot α − cot β)

h = 20 ÷ (2/√3) = 20√3/2

h = 10√3 ≈ 17.32 m

💡 Looking down instead? An angle of depression of θ from the tower top equals an angle of elevation of θ from the far point (alternate angles) — so these exact same triangles solve depression questions too.

Quick check: from the top of a m tower the angle of depression of a boat is . Since , the base equals the height, so the boat is m from the foot.

Q1From the top of a 60 m tower the angle of depression of a car is 45°. How far is the car from the base of the tower?

Q2A man walks 6 km east, then 8 km north. How far is he from his starting point?

Q3Two points on the same side of a tower give angles of elevation of 30° and 60°. Which point is closer to the tower?

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Real exam questions — Height & Distance Advanced

7 question types · 21 solved examples from real SSC papers

Advanced height and distance questions stack two right triangles on a shared height, so the trick is to write a tangent for each triangle and then subtract or add the bases to eliminate the unknown.

How to solve this type

Both triangles share the same height

h

, so write

\tan\theta=\dfrac{h}{\text{base}}

for each, turn them into bases

h\cot\theta

, and SUBTRACT (same side) to kill the common height. The standard result for distance

d

between the observers is

h=\dfrac{d}{\cot\alpha-\cot\beta}

. Never solve the two triangles separately.

A vertical tower stands on level ground. From a point P, $78$ m from the base, the angle of elevation of the top is $30^\circ$ . The tower is then raised so the elevation from P becomes $60^\circ$ . By how much was the tower raised?

26\sqrt3

40\sqrt3

52\sqrt3

78\sqrt3

The angle of elevation of a chimney top is $30^\circ$ from a point $240$ m from its base, and $60^\circ$ from a nearer point. How far is the nearer point from the base?

60

80

100

120

P and Q are two points due north of a pole, with $PQ=30$ m and Q nearer the pole. The elevation of the top is $45^\circ$ from P and $60^\circ$ from Q. Find the height of the pole.

30\sqrt3

15(\sqrt3-1)

15(3+\sqrt3)

15(3-\sqrt3)