Triangular prism — Cube (10 pieces)
Pentagonal prism — Cube (10 pieces)
Hexagonal prism — Cube (8 pieces)
Starting with the prism, this dissection can be performed with just five straight cuts.
The dissection is translational.
Heptagonal prism — Cube (11 pieces)
Octagonal prism — Cube (10 pieces)
Unfortunately the shaded piece is very thin in the middle, so there is room for improvement here.
Truncated octahedron — Cube (11 pieces)
Starting with the truncated octahedron, this dissection can be performed with just four straight cuts.
The dissection is translational.
Rhombic dodecahedron — Cube (11 pieces)
Stellated rhombic dodecahedron — Cube (12 pieces)
Truncated octahedron — Rhombic dodecahedron (24 pieces)
Stellated rhombic dodecahedron — Truncated octahedron (6 pieces)
The dissection is translational and hingeable.
Stellated rhombic dodecahedron — Rhombic dodecahedron (24 pieces)
Cube — 2 Tetrahedra + Octahedron (15 pieces)
Starting with the cube, this dissection can be performed with just five straight cuts.
The dissection is translational.Rhombic triacontahedron — Cube (29 pieces)
We start by dividing the rhombic triacontahedron into 5 pieces. If it has a volume of 1 then the five pieces have volumes of 0.4, 0.2, 0.2, 0.1 and 0.1.
The four small pieces with volumes of 0.2, 0.2, 0.1 and 0.1 are rearranged into a single block.
The first cut adds 4 pieces. These pieces are rearranged. The second cut adds a further 5 pieces. Again rearrange the pieces into a clean parallel sided block. A final two parallel cuts are made adding a single piece and 5 more pieces. The pieces are now rearranged to give a final block of dimensions 0.6×1×1. The pieces required are 4+4+5+(1+5) for a total of 19 pieces.
Here are these 19 pieces shown separately and grouped for the original 4 pieces.
The central block must now be dissected. We start by making 3 parallel cuts making a total of 4 pieces. These are rearranged. Now another cut is made adding 3 more pieces. Again rearrange the pieces making a clean parallel sided block. A final cut is made adding a further 3 pieces. The pieces are now rearranged to give a final block of dimensions 0.4×1×1. The pieces required are 4+3+3 for a total of 10 pieces.
The two blocks of 0.4×1×1 and 0.6×1×1 can now be combined to make a cube with a total of 29 pieces
Here are these 10 pieces shown separately.