Four solids of the same volume, the regular tetrahedron with side [math] \frac 1 {\sqrt {2}} [/math], a quarter-octahedron tetrahedron made up of two equilateral triangles with side [math] \frac 1 {\sqrt {2}} [/math] 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 [math] \frac 12 [/math] placed on the top of the corner, and the pyramid with a square base of side [math] \frac 12 [/math] and height of the same length, the apex above a corner.[br][br]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.