Finding a height

A tower you cannot climb

How tall is the tower of your school? You cannot hold a measuring tape up to its top. But you can stand away from it, measure the distance on the ground, and measure the angle of elevation of the top.

The ground, the tower and the line of sight make a right triangle.

Seen from the angle $\theta$ at the observer, the tower height $h$ is opposite and the ground distance $d$ is adjacent. So

A worked example. Ananya stands $20\ \text{m}$ from the foot of a tower on level ground. The angle of elevation of the top is $60°$. Find the height of the tower.

1. Draw and label. The ground distance is $20\ \text{m}$, the angle is $60°$ and the height $h$ is unknown.
2. Choose the ratio. The height is opposite and the distance is adjacent, so use $\tan$.
3. Write the relation. $\tan 60° = \dfrac{h}{20}$.
4. Solve. $h = 20 \tan 60° = 20\sqrt{3} \approx 34.64\ \text{m}$.

What the sliders show

Keep the angle fixed and increase the distance. The tower gets taller. Keep the distance fixed and increase the angle. The tower again gets taller, and quickly so, because $\tan$ grows fast above $45°$.

At $45°$ the tangent is $1$, so the height equals the distance. That is a useful check for any answer.

When the eye is above the ground

A real observer does not look from the ground. Her eye is above it. The triangle then starts at the eye level, and the height we find is only the part of the tower above her eye.

A worked example. Meena's eyes are $1.6\ \text{m}$ above the ground. She stands $22\ \text{m}$ from a tree and sees its top at an angle of elevation of $60°$. How tall is the tree?

1. The part above her eye. $22 \tan 60° = 22\sqrt{3} \approx 38.11\ \text{m}$.
2. Add the eye height. The tree rises from the ground, so add $1.6\ \text{m}$.
3. The whole tree. $38.11 + 1.6 = 39.71\ \text{m}$.

If a problem gives no eye height, take the observer's eye to be on the ground.

A slanting length: the kite string

Sometimes you know the slanting length instead of the ground distance. A kite flies on a taut string. The string is the hypotenuse and the height of the kite is opposite the angle at the ground. So we use the sine:

A worked example. A taut kite string $60\ \text{m}$ long makes $45°$ with the level ground. How high is the kite above the ground?

1. Choose the ratio. The height is opposite and the string is the hypotenuse, so use $\sin$.
2. Solve. $h = 60 \sin 45° = \dfrac{60}{\sqrt{2}} \approx 42.43\ \text{m}$.

Cosine would give the distance along the ground, $60 \cos 45°$. That is a different length.

Practise with new numbers

Try it in Python

This program finds the height of a tower seen from $30\ \text{m}$ away at three angles of elevation, then checks one answer by hand.

Common mistakes

• Using $\sin$ for a ground distance. With the height and the distance, use $\tan$. Use $\sin$ only when the slanting length is given.
• Dividing instead of multiplying. The height is $d \tan \theta$, not $d \div \tan \theta$.
• Forgetting the eye height. The triangle gives only the part above the eye. Add the height of the eye.
• A calculator in radians. Set it to degrees, or check that $\tan 45°$ shows $1$.

Remember these

• Height from a ground distance: $h = d \tan \theta$.
• Height from a slanting length: $h = \text{length} \times \sin \theta$.
• If the eye is above the ground, add the eye height at the end.
• At $45°$, the height equals the ground distance.

Check yourself

Try each one on your own first. If you are stuck, open a hint. Then check with Show the answer.

1. From a point $30\ \text{m}$ from the foot of a pole, the angle of elevation of its top is $45°$. How high is the pole?

2. A kite string is $50\ \text{m}$ long and taut. It makes $30°$ with the ground. How high is the kite?

3. Rohan's eyes are $1.5\ \text{m}$ above the ground. The part of a tree above his eyes is $9\ \text{m}$ tall. How tall is the whole tree?

4. Tia says a tower seen at $60°$ from $10\ \text{m}$ is shorter than the same tower seen at $30°$ from $10\ \text{m}$. Is she right?

5. You know the ground distance and the angle of elevation. You need the height. Do you use $\sin$ or $\tan$?