Platonic solid

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A Platonic solid is a solid figure in which all faces are congruent regular polygons. There are five such solids: the cube, the regular tetrahedron, the regular octahedron, the regular dodecahedron, and the regular icosahedron.

The cube has six faces, all of which are squares. It also has eight vertices and twelve edges.

The regular tetrahedron has four faces, all of which are equilateral triangles. It also has four vertices and four edges.

The regular octahedron has eight faces, all of which are equilateral triangles. It also has six vertices and twelve edges.

The regular dodecahedron has twelve faces, all of which are regular pentagons. It also has twenty vertices and thirty edges.

The regular icosahedron has twenty faces, all of which are equilateral triangles. It also has twelve vertices and thirty edges.

Obviously, all five of them satisfy Euler's polyhedral formula.

Those are the only possible Platonic solids; it has been proved that there are no more.

Interestingly enough, if you take a Platonic solid and connect the centers of adjacent faces with line segments, those segments are the faces of another Platonic solid.

Doing this to a regular tetrahedron gives you a smaller regular tetrahedron.

Doing this to a cube gives you a regular octahedron.

Doing this to a regular octahedron gives you a cube.

Doing this to a regular dodecahedron gives you a regular icosahedron.

Doing this to a regular icosahedron gives you a regular dodecahedron.