You can cut a cube in half to reveal a regular hexagon (drag with your mouse to move it):
Here's another look at that half cube made solid - quite a surprise to find a hexagon at the heart of a cube, isn't it?:
If you'd like to make this half-cube from paper there is a net here
If you look directly at a corner of the cube, you'll notice that the outline of the cube is a regular hexagon too. Try it with the figure below - line up points p and q and you'll see the hexagonal outline formed by the blue edges:
This leads to the familiar optical illusion, where you're not sure if you are seeing vertical or horizontal blue faces of cubes. Stare at a corner to get them to change - or imagine that the light source on the top left side then on the top right. In either case, your brain is lying - you're looking at hexagons of course:
We can lop a corner off a cube, which gives us an equilateral triangle:
In fact, lopping off four corners we find a tetrahedron inside the cube. This cube is not as square as it thinks it is:
This gives us an easy way to find the volume of a tetrahedron. We take the volume of the cube and subtract the volume of the four identical pyramids that surround the tetrahedron. (I discovered this method myself, but presumably it's well known). We'll work out the volume of a pyramid first, then calculate the volume of the tetrahedron
To find out the volume of a pyramid, we can again look at a cube, this time divided into three equal portions:
Here's a look at one of those thirds. It's a pyramid:
The cube's volume is the area of its base times its height. This pyramid's volume is therefore one third of that. This is a general formula for the volume of a pyramid, as any change you make to the area of the base is reflected proportionately in the volume, and the same for the height, and shearing parallel to the base does not affect the volume.
The volume of a pyramid is one third the area of its base times its height.
Back to our tetrahedron volume calculation... If the tetrahedron has a unit side,
it is sitting in a cube of side length ,
base area and volume .
The tetrahedron's volume is the volume of the cube, less the volume of the four surrounding pyramids, ie:
(Java applet from LiveGraphics3D)
page date: 28Nov04. I enjoy correspondence stimulated by this site. You can contact me here.