Cuboctahedron

What is the volume of the cuboctahedron?

cuboctahedron
Solution
A single simplex has a base triangle with an area of $$A_{base} =\frac{1}{2}bh = \frac{1}{2}\times \frac{1}{2}\times \frac{1}{2} = \frac{1}{8}$$ The volume of each simplex is given by $$V_S= \frac{1}{3}h_{perp}\times A_{base} = \frac{1}{3}\frac{1}{2}\frac{1}{8} = \frac{1}{48}$$  The volume of the cuboctahedron is therefore $$ V_C = 1- 8V_S = 1 -\frac{1}{6} = \frac{5}{6} $$