Platonic Solids Explorer
Only five convex solids have identical regular faces meeting the same way at every vertex: tetrahedron, cube, octahedron, dodecahedron, icosahedron. This explorer renders each in rotating 3D with vertices, edges and translucent faces, counts V, E and F to verify Euler’s formula V − E + F = 2, and highlights dual pairs — the cube and octahedron, the dodecahedron and icosahedron — whose roles of faces and vertices swap.
Runs 100% in your browser — simulations are computed locally on your device.
Read the full guide to this tool
Notes
- Euler’s formula V − E + F = 2 holds for every convex polyhedron, not only the Platonic five.
- The proof that only five exist: the angles meeting at a vertex must sum to less than 360°.
- Duals swap vertices and faces: the octahedron is the cube with face-centres promoted to vertices.
- Runs 100% in your browser — simulations are computed locally on your device.
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