Platonic Solids Calculator (Tetrahedron, Cube, Octahedron, Dodecahedron, Icosahedron)
Volume, surface area, the three radii (corner, edge, face) and the angle between faces of any of the five regular solids, from the edge length.
Results
What this tool does
There are only five solids whose faces are all the same regular polygon, with the same number of faces meeting at every corner — the Platonic solids, known since antiquity. Volume grows with the cube of the edge and area with its square, so doubling the edge gives eight times the volume. Each solid fits between three spheres: one through all the corners, one touching every edge and one touching every face, and the dodecahedron and icosahedron are the only two whose measures involve √5, the golden ratio's square root. Use the same unit for the edge and read the results in that unit, squared or cubed.
Formula
V = k_V a³ · A = k_A a² · tetrahedron V = a³√2 ÷ 12 · cube 6a² · icosahedron V = 5a³(3 + √5) ÷ 12
Variables
| Symbol | Meaning | Unit |
|---|---|---|
f | Regular solid | — |
a | Edge length | — |
V | Volume | — |
A | Surface area | — |
R | Circumscribed sphere radius (through the corners) | — |
RI | Inscribed sphere radius (touching the faces) | — |
RM | Mid-sphere radius (touching the edges) | — |
DI | Angle between adjacent faces | ° |
FA | Faces of the solid | — |
ED | Edges | — |
VE | Corners (vertices) | — |
Worked example
- Regular solid12
- Edge length10
- Volume7663.1190
- Surface area2064.5729
- Circumscribed sphere radius (through the corners)14.0126
- Inscribed sphere radius (touching the faces)11.1352
- Mid-sphere radius (touching the edges)13.0902
- Angle between adjacent faces116.57 °
- Faces of the solid12
- Edges30
- Corners (vertices)20
Limitations
- Mixing units is the most common source of error. Convert every input to the units shown next to each field before calculating.
- The calculation runs at full precision and only the display is rounded. If you copy an intermediate value and retype it, small differences can appear.