Three tetrahedra and a pyramid
- Author:
- Christian Mercat
Four solids of the same volume, the regular tetrahedron with side , a quarter-octahedron tetrahedron made up of two equilateral triangles with side and a square folded at right angles according to the hypotenuse unit, and the twenty-fourth of a cube, composed of a half square of unit hypotenuse, of a height placed on the top of the corner, and the pyramid with a square base of side and height of the same length, the apex above a corner.
The third cube pyramid unfolds into the twenty-fourth cube, which, by a principle of Cavalieri sitting on the half-square, turns into the quarter octahedron. Then, by changing the base on an equilateral triangle, it regularizes into a regular tetrahedron.
Analytically prove the equality of the volumes of these solids.