Of course $\pi$ shows up in the circumference of a circle of radius $1$, but why does it come up in the surface area of a $2$-sphere of radius $1$? One relation between the two is the remarkable fact observed by Archimedes that the horizontal projection from a sphere to a circumscribing cylinder preserves area.

I'd like to see a similar explanation for the higher powers of $\pi$ in the surface measures of higher dimensional spheres.