Problem:
A regular tetrahedron has a square shadow of area when projected onto a flat surface (light is shone perpendicular onto the plane). Compute the sidelength of the regular tetrahedron.
(For example, the shadow of a sphere with radius onto a flat surface is a disk of radius .)
Solution:
Imagine the shadow of the skeleton of the tetrahedron (i.e. make the entire tetrahedron translucent except for the edges). The diagonals of the square shadow must correspond to a pair of opposite edges of the tetrahedron. Both of these edges must be parallel to the plane—if they weren't, then edges corresponding to the four sides of the square would have to have different lengths. Thus, the length of a diagonal of the square (namely, ) must be the same as the edge length of the tetrahedron.