How do you calculate the length of a planets shadow?

Question

Given the diameter of the sun, the diameter of a planet, and the distance between them, how do you calculate the length of the planets shadow?

Answer 1

By similar triangles. Call the sun's radius R, the planet's radius r, the sun-planet distance d, and the planet's umbra's length u.

Then the ratio R : d+u is the same as r : u. Rearrange to isolate u, and Bob's your uncle.