Try and Solve: Geometry Problem

Question

Consider a regular tetrahedron, T with side length 2a. At each corner, slice off a regular tetrahedron of side length a. What is left is an octahedron, P with side lengths a. Use this dissection to determine the volume of P in terms of a.

Answer 1

Volume of tetrahedron is a³ / (6√2)

The volume of each removed tetrahedron is 1/4 of that (by volumes of similar solids)

So 3/4 of the original volume is removed leaving (1/4)(a³ / (6√2).

QED