Donatello starts with a marble cube. He then slices a pyramid off each corner, so that in the resulting polyhedron, all the edges have the same side length. If the side length of the original cube is $6$, then find the volume of the resulting polyhedron.

tomtom Dec 26, 2023

#1**+1 **

When Donatello slices off the corners, the vertices of the resulting polyhedron are the midpoints of the original cube's edges.

Donatello makes six of these slices, so our polyhedron has $\frac{2}{3}$ of the volume of the original cube.

The resulting polyhedron thus has volume $\frac{2}{3} \cdot 6^3=\boxed{288}$.

BuiIderBoi Dec 26, 2023