everyday
did you know you can estimate a building’s height using its shadow?
the sun hits you and the building at the same angle, so your height and shadow are a portable measuring stick.
1.8 m × (14 ÷ 1.2) ≈ 21 msame sun, same angle, same ratio — the building just scaled up.
the idea
stand next to a building on a sunny day and you own a measuring device. the sun hits everything at the same angle, so the ratio of your height to your shadow is the same as the building’s.
how it works
measure your height and your shadow. their ratio is the sun’s angle written as a number.
go deeper
shadows keep the same proportions
if you are 1.8 m tall and cast a 1.2 m shadow, everything around you is casting a shadow 1.2 ÷ 1.8 — two thirds — of its height.
measure the building’s shadow, divide by that same fraction, and you have its height. no ladder required.
one ratio, applied twice
measure your height and your shadow. their ratio is the sun’s angle written as a number.
the building obeys the same ratio. multiply its shadow length by your height-to-shadow ratio and out comes the estimate.
building height = your height × (building shadow ÷ your shadow)
similar triangles
you, your shadow and the sun’s ray form a right triangle. the building, its shadow and the same ray form a bigger one with identical angles.
triangles with identical angles are called similar, and their sides scale by the same factor. that factor is the whole trick.
this is possibly the oldest applied maths we know — thales of miletus is said to have measured the pyramids this way around 600 bce.
h₁ ÷ s₁ = h₂ ÷ s₂
things to wonder about next
- early morning and late afternoon give long shadows — easier to measure, but watch out for sloping ground.
- no sun? a mirror on the ground works too: sight the top of the building in it and the same triangles appear.
- surveyors and astronomers still use versions of this idea, just with far better kit.
sources & further reading
concepts: similar triangles · ratios · proportional reasoning
3 minute read