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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.
Beginner
2
Instructions
1
1
Understand the principle
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.
2
2
Set up a vertical staff
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.
Materials for this step:
Dowel Rod1 pieceTools needed:
Cotton Kitchen String3
3
Measure the staff's shadow
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.
Tools needed:
Cotton Kitchen String
Chalk Line4
4
Use the simplest moment (Thales' trick)
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.
5
5
Measure the tall object's 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.
Tools needed:
Cotton Kitchen String6
6
Apply the proportion
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.
7
7
Work a numerical example
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.
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8
Check on level ground
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.
Connected Blueprint Materials
You can swap these in
Can't get one of the materials? Swap it for an equivalent — these work just as well.
- Instead of Dowel Rod, try:
Pine Dowel Rod Set (12 Diameters)
Dowel Rod Assortment (12 Sizes) - Instead of Chalk Line, try:
Fishing Line (Monofilament)
Drying Line
Nylon Monofilament Line (0.3mm, 50m)
Nylon Fret Line - Instead of Cotton Kitchen String, try:
Violin String Set (Full Size, Synthetic Core)
Guitar String Set (Acoustic, Phosphor Bronze)
Bow String
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AwlFrequently used with this build's materials
Red Alder BoardUsed together and in similar builds
Sloyd Carving KnifeFrequently used with this build's materials
PVA Wood GlueFrequently used with this build's materials
HacksawFrequently used with this build's materials
Dry Softwood BoardFrequently used with this build's materials
Chemical Splash GogglesFrequently used with this build's materials
CordageFrequently used with this build's materialsRelated Blueprints
These blueprints share knowledge with this one — techniques, materials, or principles that connect them in the learning graph.
Related blueprints
Other builds that share materials, tools, or techniques with this one.
Measuring the Earth from a Single Mountain — Al-Biruni's Horizon-Dip Method
Measuring the Circumference of the Earth — Eratosthenes' Shadow Method
Carving a Slit Drum — The Hollow Log That Talks
Testing Seed Viability — The Rag-Doll Germination Test
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CC0 Public Domain
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