Math

Measure the School Without Touching It

Find the height of a building using a shadow, a meter stick, and similar triangles. Then measure it a second way with a paper clinometer and see whether the two answers agree.

Hard · 90 minutes

Introduction

You cannot climb a school building with a tape measure. You do not have to. Two thousand years ago people worked out the height of the pyramids from the ground, using nothing but a stick and the sun.

You are going to do it twice, two completely different ways. The interesting part is not either answer on its own. It is whether they agree.

The Why

The shadow method works because the sun is so far away that its rays arrive essentially parallel, so you and the building cast shadows at the same angle at the same moment. That makes two triangles the same shape at different sizes, which is what similar means, and in similar triangles the ratio of height to shadow is identical. The clinometer method uses a completely unrelated idea, the tangent of an angle. When two unrelated methods land on nearly the same number, that agreement is your evidence. When they do not, one of your measurements is wrong, and working out which one is the real skill.

Step-by-Step Instructions

1
Pick a building with a base you can walk right up to and a shadow that falls on flat, open ground. A wall standing on a slope will ruin both methods.
2
Stand up straight in the sun and have somebody mark where the top of your shadow lands. Measure your shadow, then measure your own height with your shoes on. Write both down along with the time.
3
Within the next few minutes, measure the building's shadow from the wall out to its tip. Shadows move, so if more than about ten minutes have passed, measure your own shadow again.
4
Work out the first answer: building height equals your height, multiplied by the building's shadow, divided by your shadow. Write the answer down with its units.
5
Now build the clinometer. Tape the straw along the flat edge of the protractor, tie the string at the center point, and hang the weight from it so it swings freely.
6
Walk a measured distance back from the base, sight through the straw at the very top of the building, and have your helper read the angle where the string hangs. Subtract that reading from 90 to get the angle up to the top.
7
Work out the second answer: height equals your distance from the building, multiplied by the tangent of that angle, plus the height of your eye above the ground. Compare it with the shadow answer and write the difference as a percentage of the larger one.

Two Methods, One Building

Fill in both halves before you compare them. Writing the units beside every single number is what catches most of the mistakes.

Measurement	Value	Units	Time taken
Row 1, Measurement	Row 1, Value	Row 1, Units	Row 1, Time taken
Row 2, Measurement	Row 2, Value	Row 2, Units	Row 2, Time taken
Row 3, Measurement	Row 3, Value	Row 3, Units	Row 3, Time taken
Row 4, Measurement	Row 4, Value	Row 4, Units	Row 4, Time taken
Row 5, Measurement	Row 5, Value	Row 5, Units	Row 5, Time taken
Row 6, Measurement	Row 6, Value	Row 6, Units	Row 6, Time taken
Row 7, Measurement	Row 7, Value	Row 7, Units	Row 7, Time taken
Row 8, Measurement	Row 8, Value	Row 8, Units	Row 8, Time taken
Row 9, Measurement	Row 9, Value	Row 9, Units	Row 9, Time taken
Row 10, Measurement	Row 10, Value	Row 10, Units	Row 10, Time taken

Rows to fill: your height, your shadow, the building's shadow, the shadow-method answer, your distance from the base, the angle you read, your eye height, the tangent-method answer, the difference, and that difference as a percentage.

Materials List

A sunny day, ideally mid-morning or mid-afternoon when shadows are long and sharp
A meter stick or a tape measure
A friend or an adult to help hold and mark
Chalk or small stones to mark where a shadow ends
A protractor, a drinking straw, string, and a small weight such as a washer, for the clinometer
A calculator with a tan button, for the second method

Safety First!

Stay on the sidewalk or on grounds you are allowed to be on, and ask permission before measuring around a building that is not yours. Never look at the sun, including through the straw. Watch for traffic while pacing out distances, and do the pacing with an adult if the ground runs anywhere near a road.

Challenge Mode

Get the two answers within five percent of each other. If they are further apart than that, hunt down the cause instead of averaging them: sloping ground, a shadow measured too many minutes later, an eye height you forgot to add, or a roof that overhangs the wall. Then run the whole method on something whose height you can look up, like a basketball hoop at ten feet, and find out how close your version really is.