Question: Proving the volume of a pyramid

[Jehanne]

Yup. If you draw a line segment from alternating vertices of a hexagon to the center, it will look like a cube with a vertex facing you.

Like a box that got stepped on?

[Fireball]

Like this-

[The Valkyrie]

Like this-

It's too early in the morning to be making me think.

Stop it!

[polymath257]

Like this-

Actually, that is slightly different from what I was describing, but still very nice. The vertices of your hexagon are not all on the same plane.

This is, actually, a good way to see the decomposition of the cube into three pyramids. The opposite vertex (that you can't see in the projection) connects to each of the visible square sides to form a pyramid. So the volume of a pyramid is 1/3 the volume of the cube.

[Fireball]

Like this-

Actually, that is slightly different from what I was describing, but still very nice. The vertices of your hexagon are not all on the same plane.

This is, actually, a good way to see the decomposition of the cube into three pyramids.  The opposite vertex (that you can't see in the projection) connects to each of the visible square sides to form a pyramid. So the volume of a pyramid is 1/3 the volume of the cube.

Oy. Now I think on it, I realize what you meant. -Hexagon inscribed in a cube

[Anomalocaris]

The geometric demonstration of 6 identical square based pyramids with height equal to half of the sides fitting into a cube does not seem generalizable to pyramid with non-square bases or other height to base ratios.

[Jehanne]

Oy. Now I think on it, I realize what you meant. -Hexagon inscribed in a cube

And to think, per some archeologists, that it took Humanity 10,000 years after the invention of the pottery wheel for someone to come along and rotate it by 90 degrees.

[polymath257]

The geometric demonstration of 6 identical square based pyramids with height equal to half of the sides fitting into a cube does not seem generalizable to pyramid with non-square bases or other height to base ratios.

For that, we need more general facts about what happens under expansion or contraction along different directions.

Or, you could compare corresponding rectangular figures approximating the two pyramids and compare the volumes of the approximations. This is close to what Archimedes would have done. It is *close* to calculus, but not there yet.