Interactive Shadow Calculator
The Legend
Thales of Miletus (c. 624-546 BCE), one of the Seven Sages of Greece, allegedly measured the Great Pyramid of Giza during his travels in Egypt. The Egyptian priests were amazed when he calculated the pyramid's height without climbing it. Thales waited for the moment when his own shadow equaled his height (sun at 45°), then measured the pyramid's shadow. Alternatively, he used proportions: the ratio of any object's height to its shadow equals the ratio for all objects at the same moment.
The Mathematics
Similar Triangles: When the sun shines at angle θ, every vertical object and its shadow form a right triangle with the same angle. Therefore, these triangles are similar.
Proportion:
h_pyramid / s_pyramid = h_stick / s_stick
Where h is height and s is shadow length.
Solving for pyramid height:
h_pyramid = s_pyramid × (h_stick / s_stick)
Special Case (θ = 45°):
When the sun is at 45°, tan(45°) = 1, so shadow length equals object height. Thales could simply measure the pyramid's shadow at that moment!
Tangent Relation:
tan(θ) = height / shadow
height = shadow × tan(θ)