Maths shelf · Class 10

How tall is the flagpole? Ask its shadow

A 1.5 m stick casts a 2 m shadow. At the same moment, the school flagpole casts an 8 m shadow. How tall is the flagpole, without climbing it?

About 40 minutes, outdoors on a sunny day
Class 10
Mathematics and money

What you will learn: Use shadows measured at the same moment and similar triangles to find a height you cannot reach, and explain what makes the method fair.

What should I know first?

If this feels new, try this first. It is fine to go back.

Dadi's chai for six Class 7

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

Never look at the Sun, even for a moment. Measure the shadows on the ground.

Keep it safe:

Never look directly at the Sun, with or without sunglasses: look only at the shadows on the ground.
Work on open, level ground away from traffic. Hold sticks upright and away from faces. Have water and a hat on hot days.
If you weigh down the stick with stones or use pegs to mark shadow tips, keep them away from babies: nothing goes in the mouth.

You will need

A straight stick or rulerAbout 1 m to 1.5 m long.
A measuring tapeOr measure your stride and pace the shadows.
A friend, paper and a pencilOne holds the stick straight; one measures.

Time: About 40 minutes, outdoors on a sunny day.

What do you think?

The pole's shadow is 4 times as long as the stick's. Is the pole 4 times as tall as the stick?

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, always

Write your reason. Then ask: would it still work if the two shadows were measured an hour apart? Test it with a stick at two times.

Yes, if measured at the same time

Write your reason. Why would the time matter? Measure a stick's shadow now and in an hour, and compare.

No, shadows do not work

Write your reason. Then test it: measure two sticks of different heights and their shadows at the same moment.

Not sure

That is fine. Stand two sticks of different heights side by side in the sun. Measure each stick and its shadow.

Need a hint?

At the same moment, height ÷ shadow is the same for every upright object nearby.

The story

Before Independence Day, the school needs a new rope for the flagpole. "How long must it be?" asks the teacher. Nobody can climb up to measure. Tanmay holds a 1.5 m stick straight up; its shadow is 2 m long. Tara paces the pole's shadow: 8 m. "The pole's shadow is four times as long," says Tara. "So is the pole four times as tall?" Dhruv frowns. "Only if we measured both at the same time."

The idea

Two triangles are similar if their angles are equal. Then their matching sides are in the same ratio.

At one moment, the Sun's rays make the same angle with level ground everywhere nearby. An upright stick and an upright pole make similar triangles with their shadows.

So height ÷ shadow is the same for both: 1.5 ÷ 2 = 0.75, and pole = 0.75 × 8 = 6 m.

The method needs: both measured at the same time, upright objects, level ground, and the shadow measured from the foot of the object.

Later in the day the Sun is lower and shadows are longer, so the ratio changes. That is why the stick must be measured at the same time as the pole.

A stick 1.5 m tall with a 2 m shadow, and a flagpole of unknown height with an 8 m shadow. Dashed lines show parallel sunrays from the top of each to the end of its shadow, making two similar triangles.

Worked example

The flagpole

Stick 1.5 m, shadow 2 m. Flagpole shadow 8 m at the same moment. Find the height.

Set up the ratio: pole ÷ 8 = 1.5 ÷ 2.
Multiply both sides by 8: pole = 1.5 × 8 ÷ 2.
pole = 12 ÷ 2 = 6.

The flagpole is about 6 m tall, so the rope must be about 12 m (up and down) plus extra for tying.

Try it

Guess first.
Look at a tall object you cannot reach. Write your guess of its height.
Measure the stick.
Hold the stick upright on level ground. Measure its height and its shadow, from the foot of the stick.
Measure the object's shadow.
At once, measure the shadow of the tall object, from its foot to the tip of the shadow.
Work it out.
Height = stick height × object shadow ÷ stick shadow. Compare with your guess.
Check it.
Repeat with a different stick, still at the same time. Do you get nearly the same height?
Find the weak points.
Write two things that could make your answer wrong, and how you would reduce them.

What you make (paper or offline)

Your shadow table

A table: the stick is 1.5 m tall with a 2 m shadow, ratio 0.75; the flagpole has an 8 m shadow and the same ratio 0.75, so its height is 6 m. A blank row for your own measurement.

Record each object, its height (if known), its shadow and the ratio.

Use the ratio to find the unknown height.

Write the time you measured, and the weak points.

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.

The Sun is so far away that its rays arrive almost parallel. At the same moment, they meet every upright object at the same angle.

So the stick, its shadow and its ray make a triangle similar to the pole, its shadow and its ray. Their sides are in the same ratio.

Height of pole = 1.5 × 8 ÷ 2 = 6 m.

Think back

What makes the stick's triangle and the pole's triangle similar?

When would the angle method (with a protractor) be better than shadows?

Investigate further

Measure the same stick's shadow every hour from 9 am to 3 pm. Graph the shadow length. When is it shortest?
Use your stride as a measuring unit. How could you check how long your stride is?

Can you use it here?

A 1 m stick casts a 0.8 m shadow. A tree casts a 12 m shadow at the same moment. How tall is the tree?

Ask someone at home

Measure the shadow of a family member and of a lamp post or wall at the same moment. Estimate the post's height and ask someone to check your working.

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 sunny day, level open ground away from traffic, a stick and a tape measure. Remind children never to look at the Sun.
Water and hats on hot days.

Questions that help

"Why the same moment?"
"Which numbers go on top of the fraction?"
"How could your answer be wrong?"

What's really going on

At one moment, the Sun's rays are parallel, so an upright stick and pole form similar triangles with their shadows.

Matching sides of similar triangles are in the same ratio: height ÷ shadow is the same for both.

Good measurement means naming weak points: time, level ground, upright objects, where the shadow starts.

Other ways to do it

Cloudy day? Use the numbers on this page.
Use a torch and toy figures indoors to make shadows on a table.

For teachersLearning aims, classroom ideas and what to look for.

Learning aims

Use similarity of triangles to find an inaccessible height.
Set up and solve a proportion.
Identify conditions and sources of error in an indirect measurement.

Subject links

Mathematics: similar triangles and proportion.
Class 10 "How tall is the school building?" (angles and tan).
Class 7 "Dadi's chai for six" (ratio and scaling).

These links are general. They have not been checked against a specific board syllabus.

In the classroom

Groups measure the same object's shadow with different sticks at the same time and compare heights.
Compare the shadow method with the protractor method for the same building.

What to look for

Puts the ratio the right way up.
Insists on the same moment.
Names at least two sources of error.