Platonic Solids Calculator

The five Platonic solids are the only convex polyhedra where every face is the same regular polygon and every vertex looks like every other. There are exactly five, and the proof is short: the angles meeting at a corner must add to less than 360°, which permits only three, four or five triangles, three squares, or three pentagons. Six triangles or four squares lie flat, and three hexagons already fill the full 360°, so the list closes there.

Plato tied them to the classical elements. That part did not survive, but the mathematics did.

Each solid has exact closed-form formulas for its surface area, volume, and all three radii: the inradius (the sphere that touches the middle of every face), the midradius (the sphere tangent to every edge), and the circumradius (the sphere through every vertex).

All five, in terms of edge length a:

And the three radii, as multiples of a:

Euler’s formula V − E + F = 2 holds for all five, and for any convex polyhedron. Check it against the table: 4 − 6 + 4, 8 − 12 + 6, 6 − 12 + 8, 20 − 30 + 12, 12 − 30 + 20. All 2.

Which one is roundest? Not the biggest. For a fixed edge the dodecahedron is far and away the largest solid, but sphericity ranks them icosahedron 0.939, dodecahedron 0.910, octahedron 0.846, cube 0.806, tetrahedron 0.671. Size and roundness are different questions, and the dodecahedron is the standing proof of it.

The dodecahedron and icosahedron are duals: put a vertex at the centre of each face of one and you get the other. The cube and octahedron are duals too, which is why their face and vertex counts swap. The tetrahedron is its own dual.

In chemistry the icosahedron turns up in boron clusters and viral capsids, the cube in simple cubic lattices, and the tetrahedron in carbon bonding, where sp³ hybridisation gives bond angles of 109.47°.