Maths shelf · Class 10
How tall is the school building?
You cannot climb the school building to measure it. Can a protractor, a straw and a measuring tape on the ground find its height?
What you will learn: Use tan to find a height from a horizontal distance and an angle of elevation, add the instrument's height, and show how a small error in the angle changes the answer.
What should I know first?
If this feels new, try these first. It is fine to go back.
- Map the fair Class 9
- Design an accessible ramp Class 9
Small test at the end: scroll down to the end of this page for five short questions that check what you have learnt. Tap an answer, then press Check answer. Go to the questions ↓
Jump to: Check yourself · For grown-ups · For teachers
Safety first
Stay on the ground, away from roads, with a grown-up. Never look at the Sun.
Keep it safe:
- Stay on level ground in the school or a park, away from roads and traffic. No climbing, ever.
- Never look at the Sun through the straw or at all.
- Keep the weight on the thread (a washer or bead) away from babies: nothing goes in the mouth.
You will need
- A protractor, a straw, thread and a small weightTo make a clinometer.
- A long measuring tapeFor the distance along the ground.
- A calculator or tan tableFor tan of the angle.
Time: About 45 minutes.
What do you think?
Tara stands 20 m from the wall and sees the top at 35°. Tanmay says: "tan 35° is about 0.7, so the building is 14 m." Is he right?
Make a prediction and write one reason for it. Then tap the guess closest to yours to see how to test it. It is fine to change your mind after checking.
Yes, 14 m
Where is Tara's eye? The triangle starts there, not at the ground.
Almost, add something
What do you need to add, and why?
No, it is 20 m
At one angle, the height above your eye equals the distance. Which angle? And what would you still need to add?
Not sure
That is fine. Draw the triangle from Tara's eye to the top of the building, and label what you know.
Need a hint?
Draw the triangle. Its bottom is level with Tara's eye, 1.5 m above the ground.
The story
The school wants a banner from the roof to the ground, and nobody knows how tall the building is. "We could climb up with a tape," says Tanmay. "No climbing," says Dhruv firmly. Tara makes a clinometer from a protractor, a straw and a thread with a weight, and stands 20 m from the wall on the level playground. The thread crosses the protractor at 55°, so the angle of elevation is 90° − 55° = 35°. "Now it is a triangle problem," she says.
The idea
You met rise ÷ run on the Class 9 ramp: a 1:12 ramp rises 1 cm for every 12 cm across. Rise ÷ run is the gradient, and tan of the slope angle is that same number. Here the rise is the height above your eye and the run is your distance from the wall.
In a right-angled triangle, tan of an angle = opposite side ÷ adjacent side.
The angle of elevation is between the level line from your eye and your line of sight to the top.
The adjacent side must be the horizontal distance. A distance measured up or down a slope is the wrong side of the triangle.
The triangle starts at your eye, so add your eye height (or the clinometer's height) at the end.
Small errors in the angle grow with distance. Repeat the reading and give a range of estimates, not one exact number. The range only covers the angle; a wrong distance or eye height adds more error.
Worked example
The tree at 45°
Dhruv stands 12 m from a tree on level ground. With his clinometer at 1.5 m, the top is at 45°. How tall is the tree?
- tan 45° = 1, so the height above the clinometer = 12 × 1 = 12 m.
- Add the clinometer height: 12 + 1.5 = 13.5 m.
- Check: at 45° the triangle has two equal sides, so 12 m up for 12 m across makes sense.
The tree is about 13.5 m tall.
Try it
Make a clinometer.
Tape a straw along the flat edge of a protractor. Hang a thread with a small weight from the centre hole.
Measure the distance.
On level ground, measure the horizontal distance from the wall to where you stand. Write it down.
Stop walking before you look up. Lower the clinometer before you move again.
Read the angle.
Look at the top of the wall through the straw. A partner reads where the thread crosses the protractor. The angle of elevation is the difference between that reading and 90°: a reading of 55° (or 125°) means 35°.
Measure your eye height.
Measure from the ground to the straw.
Work it out.
Height = distance × tan(angle) + eye height.
Give a range.
Take three readings. Work out the height for the smallest and largest angle, and give the answer as a range of estimates.
What you make (paper or offline)
Your height card
Make a height card for a building or tree, with your distance, eye height and three angle readings.
Work out the height for each reading.
Give your answer as a range, such as "about 15 to 16 m".
Make it on paper. Keep it with you. Nothing to upload.
What did we find out?
Read this after you have tried it. Look back at your guess and your reason first.
tan 35° = height above the eye ÷ horizontal distance, so the height above the eye = 20 × tan 35° ≈ 20 × 0.70 = 14 m.
The triangle starts at Tara's eye, 1.5 m up, so the building is about 14 + 1.5 = 15.5 m tall.
If the reading could be 1° out (and the distance and eye height are right), the estimates run from about 15.0 m to 16.0 m. Say "about 15.5 m", not "exactly".
Check yourself
Five questions about what this page teaches. Choose an answer and press Check answer, or open "See the answer and why". Nothing is saved or sent: your choices stay on this screen and disappear when you leave or reload the page.
How do you want to answer? You choose, and you can change at any time.
See the answer and why
- A. 13.5 m Correct. tan 45° = 1, so 12 m above the clinometer, plus 1.5 m.
- B. 10.5 m Not this one. That takes the 1.5 m away. The clinometer is above the ground, so add it.
- C. 17 m Not this one. That is about the length of the sloping line of sight, not the height.
- D. 12 m Not this one. That is only the height above the clinometer. Add the 1.5 m.
See the answer and why
- A. 14 m Not this one. That is only the part above her eye.
- B. 15.5 m Correct. 20 × 0.70 = 14 m above her eye, plus 1.5 m.
- C. 21.5 m Not this one. That adds the distance and the eye height. You need tan.
- D. 28.6 m Not this one. That is 20 ÷ 0.70. Height = distance × tan, not distance ÷ tan.
See the answer and why
- A. A sloping distance is always shorter Not this one. It is longer, and it is the wrong side of the triangle.
- B. Tan uses the side at right angles to the height Correct. A sloping distance is not the adjacent side of the right-angled triangle.
- C. It does not matter for a tall building Not this one. It always matters: the wrong side gives a wrong height.
- D. Tapes measure only on flat ground Not this one. Tapes work on slopes too. The issue is which side of the triangle you measure.
See the answer and why
- A. Exactly 15.5 m Not this one. Her reading might be 1° out, so "exactly" claims too much.
- B. Somewhere between 0 and 30 m Not this one. That is true but far too wide to be useful.
- C. As a range: about 15 to 16 m Correct. 20 × tan 34° + 1.5 ≈ 14.99 m and 20 × tan 36° + 1.5 ≈ 16.03 m, if the distance and eye height are right.
- D. About 13.5 to 14.5 m Not this one. That forgets to add her 1.5 m eye height.
See the answer and why
- A. cos = adjacent ÷ hypotenuse Not this one. Cosine links the ground distance to the sloping line. Tangent links the two lengths in the question.
- B. opposite × adjacent Not this one. Multiplying the sides does not give an angle ratio.
- C. tan = opposite ÷ adjacent Correct. The height is opposite the angle; the ground distance is next to it.
- D. sin = opposite ÷ hypotenuse Not this one. Sine links the height to the sloping line of sight, which you have not measured.
Think back
Why do we add the eye height at the end?
Why is a range more honest than one exact number?
Investigate further
- Stand at two different distances from the same building. Do you get the same height? Why might it differ?
- On paper only: how could you find the width of a river without going near it? Draw the triangle.
Can you use it here?
At 45°, tan is 1, so the height above your eye equals the distance. Then add your eye height. Find a spot where the top of a tree is at 45°: how tall is it?
Ask someone at home
Ask someone at home how they would find the height of a tall tree without climbing it.
No one free to join in? That's fine. You can do it by yourself with drawings, and ask a grown-up to help with the materials.
For grown-upsTap a heading to open it. The safety notes are at the top of the page.
Before you start
- A protractor, straw, thread, a small weight and a long tape. Keep the weight away from babies; nothing in the mouth.
- Choose a school ground or park, away from roads and water. No climbing, never look at the Sun, and stop walking before looking up.
Questions that help
- "Where does the triangle start?"
- "Is that distance level?"
- "How sure are you of the angle?"
What's really going on
This is the classic "heights and distances" use of tan. The usual mistakes are forgetting the eye height and using a sloping distance.
The thread hangs vertically. The protractor measures its angle from the straw's baseline, so the angle of elevation is the difference between the thread reading and 90° (|90° − reading|).
Repeating readings and giving a range of estimates is how real surveyors work; it is not a sign of failure. The range covers only the angle error, not a wrong distance.
Other ways to do it
- No protractor? Use a phone's spirit-level app with an angle reading, with a grown-up.
- Use a tall tree or a flagpole in a park instead of the school.
For teachersLearning aims, classroom ideas and what to look for.
Learning aims
- Apply tan to find heights from angles of elevation.
- Identify the horizontal distance and add the instrument height.
- Express a result with a sensible range.
Subject links
- Maths: some applications of trigonometry, heights and distances (confirm the chapter in the current edition).
- Class 9 "Two journeys to school" (horizontal and vertical on a graph).
- Class 12 "Limitations and error".
These links are general. They have not been checked against a specific board syllabus.
In the classroom
- Groups measure the same flagpole from different distances and compare results.
- Collect every group's answer and discuss why they differ.
What to look for
- Adds the eye height.
- Uses a level distance.
- Gives a range and explains it.