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# Geometry

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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 20, 2024

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In order to calculate the area of the resulting polyhedron, we can find the volume of the original cube and then subtract the pyrmaids cut off.

First off, the area of the original cube can be represented with the equation $$V=s^3$$ where s is the sidelenght. Plugging in 6, we get

$$V=6^3=216$$

Next up, let's calculate the volume of each pyramid he sliced off.

Let's note that each pyramid removed has a BASE area of 1/4 each of the cube and the height is 1/2 the sidelength.

We know this since all the post-cutoff edges are the same length.

The volume of the cube is represented with the equation $$\frac{1}{3}bh$$ where b is the base and h is the height.

We know the base is $$b=\frac{1}{4}*6*6=9$$ and the height is $$1/2*6 = 3$$

So we have

$$1/3*9*3 = 9$$

There are 8 corners on a cube so there are 8 pyramids, so we have $$9*8=72$$

Doing the subtraction, we get $$216-72=144$$