The Platonic Solids, the Beauty of Simplicity.

The 5 platonic solids are the only polyhedra in 3 dimensions whose faces are identical regular polygons whose vertices are all identical which are convex

Vertices: 4
Edges: 6
Faces: 4
Edges per face: 3
Edges per vertex: 3
Surface area: sqrt(3) * edgelength^2
Volume: sqrt(2) / 12 * edgelength^3
Circumscribed radius: sqrt(6) / 4 * edgelength
Inscribed radius: sqrt(6) / 12 * edgelength

Vertices: 6
Edges: 12
Faces: 8
Edges per face:3
Edges per vertex: 4
Surface area: 2 * sqrt(3) * edgelength^2
Volume: sqrt(2) / 3 * edgelength^3
Circumscribed radius: sqrt(2) / 2 * edgelength
Inscribed radius: sqrt(6) / 6 * edgelength

Vertices: 8
Edges: 12
Faces: 6
Edges per face: 4
Edges per vertex: 3
Surface area: 6 * edgelength^2
Volume: edgelength^3
Circumscribed radius: sqrt(3) / 2 * edgelength
Inscribed radius: 1 / 2 * edgelength

Vertices: 12
Edges: 30
Faces: 20
Edges per face: 3
Edges per vertex: 5
Surface area: 5 * sqrt(3) * edgelength^2
Volume: 5 * sqrt(3 + sqrt(5)) / 12 * edgelength^3
Circumscribed radius: sqrt(10 + 2 * sqrt(5)) / 4 * edgelength
Inscribed radius: sqrt(42 + 18 * sqrt(5)) / 12 * edgelength

Vertices: 20
Edges: 30
Faces: 12
Edges per face: 5
Edges per vertex: 3
Surface area: 3 * sqrt(25 + 10 * sqrt(5)) * edgelength^2
Volume: sqrt(15 + 7 * sqrt(5)) / 4 * edgelength^3
Circumscribed radius: sqrt(sqrt(15) + sqrt(3)) / 4 * edgelength
Inscribed radius: sqrt(250 + 110 * sqrt(5)) / 20 * edgelength

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