Measuring the Height of a Pyramid by Its Shadow — Thales' Similar Triangles

Around 600 BC the Greek philosopher Thales of Miletus astonished the Egyptians by finding the height of the Great Pyramid without climbing it — using only its shadow and a stick. His insight was that under the same sun, every upright object and its shadow form the same shape of triangle, just at different sizes. So the ratio of a stick's height to its shadow is exactly the ratio of the pyramid's height to its shadow. This blueprint reproduces Thales' method with a vertical staff and a measuring cord — the birth of measuring the unreachable by proportion.

初学者

说明

Understand the principle

The Sun's rays strike everything at the same angle at a given moment, so a tall object and its shadow make a triangle the same shape as a short stick and its shadow — only larger. Because the triangles are similar, height ÷ shadow is the same for both. Measure the stick and its shadow, measure the object's shadow, and proportion gives the object's height.

Set up a vertical staff

Stand a straight staff upright on level ground and check it against a plumb line so it is truly vertical. Measure and record its height above the ground exactly.

此步骤所需材料：

Dowel Rod1 个

所需工具：

Cotton Kitchen String

Measure the staff's shadow

Measure the length of the staff's shadow from its base to the tip, and mark the tip with chalk so you can measure the tall object's shadow at the very same moment. Both shadows must be read together.

所需工具：

Cotton Kitchen String

Chalk Line

Use the simplest moment (Thales' trick)

The classic shortcut: wait until the staff's shadow is exactly as long as the staff is tall — the Sun is then at 45°. At that instant every vertical object's shadow equals its own height, so you can read the pyramid's height straight off its shadow.

Measure the tall object's shadow

Measure the shadow of the pyramid, tower or tree from the point directly below its top to the shadow tip. For a pyramid, add half the base width, because the shadow starts from the centre of the base, not the edge.

所需工具：

Cotton Kitchen String

Apply the proportion

Compute the height with similar triangles: object height = object shadow × (staff height ÷ staff shadow). If you used the 45° trick, staff height ÷ staff shadow = 1, so the object's height simply equals its shadow.

Work a numerical example

Say a 1 m staff casts a 2 m shadow — the ratio is 0.5. A tower whose shadow measures 30 m is therefore 30 × 0.5 = 15 m tall. The Great Pyramid, with a shadow scaled the same way, gives its own height of about 146 m.

Check on level ground

The result is exact as long as both shadows are measured at the same instant and the ground is level. Test it on a building of known height to confirm your measuring before trusting it on something you cannot reach.