Mathematics course : 6th grade

How to split a cube into 3 identical pyramids

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We shall build with cardboard and glue a pyramid fitting into a cube.

The base of the pyramid will be the bottom side of the cube, and the top of the pyramid will be one of the top corners of the cube.

We shall see that the volume of this pyramid is exactly 1/3 of the volume of the cube. Moreover, we shall reconstruct the cube with three identical pyramids like this one.

To build the pyramid, we begin with a model (see video for details).

Click on image to enlarge

We cut the sheet according to the model.

And we fold and paste to get a pyramid.

We build 3 pyramids on the same model. And we can see that they fit exactly into a cube.

Do the three pyramids fill in the cube entirely ? Or could there be a void inside ? Reasoning on the angles of the various planes, on can see that there can be no void inside. The cube is fully filled.

Therefore the volume of the initial pyramid is a third of the volume of the cube. Hence it is the product of the surface area of its base times its height divided by 3.

We can write : V = S x h x 1/3
(where V is the volume of the pyramid, S is the surface area of its base, that is one of the sides of the cube, and h the height, that is an edge of the cube).

We shall use this result to extend it to any kind of pyramid, and even to any cone.

Exercise:

1. Try to draw on a sheet of paper the cube with its three fitting pyramids inside.

Answer

The screens of the video

screen 1
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screen 11

Contact

Answer: we discover that the drawing looks terribly messy. That's why we built the pyramids with cardboard and glue.