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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.

 

 Jun 2, 2024
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We don't actually have to find the area of the polyhedron itself. 

 

We know how to find the area of the cube and the area of the pramids cut off, so we just have to subtract them. 

Each of the pyramids cut off must have a right iscocoles base with sidelengths 3 and height 3. 

 

From this, we can write

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

 

So the volume is 180. 

 

Thanks! :)

 Jun 2, 2024

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