We are given a cube of side length 2. We then slice a pyramid off each corner, as shown, so that every side length of the remaining polyhedron has the same length. Let $A$, $P$, $Q$, and $R$ be the vertices shown.
I don't know how to draw pictures on here, so I'll have to describe it in words.
Forget that it's a cube. Do it in two dimensions. Imagine looking at the cube side-on, so that what you see is a square. Cut off each corner of that square such that you end up with an equilateral octogon... a stop sign, if you will. Note that each corner that you cut off is an isoceles right triangle. If you can figure out the lengths of the sides of the triangle, you can then figure out the length of the sides of the octogon.