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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.

 

 May 1, 2024
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Each of the 8 pyramids being cut off has a right isosceles base with equal sides 3, and height 3.

Using the formula for volume of cube and volume of pyramid, we have

\(\text{Volume} = 6^3 - 8 \cdot \dfrac13 \left(\dfrac{3 \cdot 3}2\right)\cdot 3 = 180\)

 May 1, 2024

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