Shadows

Part of the Triangles section of Coddy's Geometry journey. Lesson 57 of 71.

At any one moment, the sun's rays all come in at the same angle. So every upright object and its shadow make a right triangle of the same shape. The triangle made by a pole and its shadow is similar to the triangle made by a building and its shadow. Height divided by shadow is the same for both, so the two heights are in the same ratio as the two shadows. In the figure, a 4 m pole casts a 6 m shadow, and a building beside it casts a 30 m shadow.

$Two right triangles side by side: a pole of 4 with a shadow of 6, and a building of unknown height h with a shadow of 30, the sun rays drawn dashed$

The same sun makes the same shape, so height over shadow is equal

Write the proportion and solve it:

h/4 = 30/6

h = 20

The building is 20 m tall. You found its height by measuring only along the ground.

This method was used long ago to measure the height of the pyramids. The proportion is the same kind as in the last chapter. The matching sides are height with height and shadow with shadow. The new step is to see that the two triangles are similar. They are similar because the sun's rays hit both objects at the same angle, and both objects make a right angle with the ground.