# decompositions

### decompositions of a regular tetrahedron into identical pentahedra / regular pyramids

 It isn't so obvious to assemble this very simple puzzle (on the left) despite the simplicity of the two pieces. The net of the half-tetrahedron (pentahedron) is easy to draw: a square with, attached to it, two opposite equilateral triangles, and two opposite isosceles trapezoids (strips of three equilateral triangles). Now we may also decompose each pentahedron into three regular pyramids (on the right) or into two smaller pentahedra (below). So we get more tricky puzzles with six or four pieces.

### decompositions of a cube into tetrahedra

40 tetrahedra, 8 regular (cyan) and 32 tri-right-angled (8 magenta and 24 blue) build a cube in two different ways.
 remove the 8 magenta pyramids, there is a cuboctahedron left remove the 24 blue pyramids, there is a stella octangula left one may remove the tetrahedra along the cube's edges (see the LiveGraphics3D help at the bottom of the page)

### decomposition of a parallelepiped into six tetrahedra of same volume

 the four yellow points may be moved to modify the parallelepiped (see LiveGraphics3D help at the bottom of the page)

proof (Vincent Papillon): drawing on the right
The three tetrahedra build un half parallelepiped (the second half is the grey prism).
Two tetrahedra have bases with same area (the red half parallelograms) and same hight (their bases are coplanar and they have the same fourth vertex); thus they have same volume. This also holds for the two tetrahedra with green bases. The three tetrahedra have thus same volume and likewise for the three tetrahedra which build the prism. Thus the six tetrahedra (animation on the left) have same volume.
remark: This result is also the proof that the volume of a tetrahedron is the sixth of the one of the parallelepiped with which it shares three edges. On the drawing on the right the tetrahedron with the green face (where the three common edges have a common vertex) and the one with the red face have this property; on the left there are two others.

 the same as above, with bright colors an other decomposition of the same type exercise: prove that the six tetrahedra have same volume

 home page convex polyhedra - non convex polyhedra - interesting polyhedra - related subjects November 2003updated 20-01-2016