Heights and Distances: Formulas, Shortcuts and Solved Examples

What is a height and distance problem?

You cannot measure a tower with a tape. So you measure an angle and a distance on the ground. Then you work out the height.

Every problem reduces to a right-angled triangle.

What are the angle of elevation and the angle of depression?

Angle of elevation: the angle your line of sight makes with the flat line when you look up at something.
Angle of depression: the same angle when you look down.

Both are measured from the flat (horizontal) line, never from the upright one.

What is the main formula?

tan θ = height ÷ distance along the ground, so height = distance × tan θ.

Why it works: in a right triangle, tan of an angle is the side opposite it divided by the side next to it. The tower is opposite and the ground is next to it.

When do I use sin or cos?

Use sin θ = opposite ÷ hypotenuse when a slanting length is given, such as a ladder or a rope. Use cos θ = base ÷ hypotenuse for the ground part.

Why it works: a ladder is the longest side of the triangle. Sin links it to the height. Cos links it to the base.

Which values must I learn?

sin 30° = 1/2, sin 45° = 1/√2, sin 60° = √3/2. tan 30° = 1/√3, tan 45° = 1, tan 60° = √3.

Why they work: the 45° triangle has two equal sides, so its tan is 1. The 30°-60°-90° triangle has sides in the ratio 1 : √3 : 2.

What if the observer moves?

Write one equation for each position. Both give the same height h. Then solve.

Why it works: the tower does not change. So the two expressions for h must be equal.

How does a shadow problem work?

tan of the sun angle = height of the pole ÷ length of its shadow.

A pole as tall as its shadow means tan θ = 1, so θ = 45°.

Easy, step by step: a ladder 64 m long makes 30° with the ground. How high does it touch the wall?

Step 1: the ladder is the slanting side. The height is opposite the angle.

Step 2: height = 64 × sin 30°.

Step 3: sin 30° = 1/2, so the height is 32 m.

Medium, step by step: from a point 4 m from a tower the angle of elevation of its top is 60°. Find the height.

Step 1: height = distance × tan 60°.

Step 2: tan 60° = √3.

Step 3: the height is 4√3 m.

Exam level, step by step: the angle of elevation of a tower top is 30°. After walking 8 m towards it, the angle is 60°. Find the height.

Step 1: let the height be h. At the first point, the distance is h ÷ tan 30° = h√3.

Step 2: at the second point, the distance is h ÷ tan 60° = h/√3.

Step 3: the difference is the 8 m walked. h√3 − h/√3 = 2h/√3 = 8.

Step 4: h = 4√3, so the height is 4√3 m.

Exam level, step by step: a man on a river bank sees a tree on the other bank at 60°. He moves back 36 m and sees it at 30°. Find the width of the river.

Step 1: let the width be x. At 60°, h = x√3.

Step 2: at 30° the distance is x + 36, so h = (x + 36)/√3.

Step 3: set them equal. 3x = x + 36, so 2x = 36.

Step 4: the width is 18 m.

Which traps cost marks?

Measuring the angle from the vertical. It is always from the horizontal.

Mixing up tan 30° and tan 60°. The bigger angle gives the bigger tan.

Forgetting the observer’s own height when the question gives it.

Using sin when you should use tan. Check which two sides are in the question.

Which shortcuts save time?

At 45° the height equals the distance.

For the two-position 30°-60° type, the height is (walk ÷ 2) × √3.

Draw a figure first, and mark the angle and the sides.

The angle of depression from the top equals the angle of elevation from the ground.

What patterns come up in exams?

Five patterns repeat. A ladder, a tower at one angle, a tower at two angles, a river width, and a shadow.

For each, decide which two sides the question gives.

Then pick sin, cos or tan to match them.