Regular Polyhedron Volume Calculator (Platonic Solids)

Find the volume of the five Platonic solids from one edge length. Tetrahedron, cube, octahedron, dodecahedron and icosahedron, with surface area, circumscribed and inscribed sphere radii, and face vertex edge counts.

Input

Choose a type of regular polyhedron and enter the edge length a to find its volume, surface area, circumscribed and inscribed sphere radii, and the number of faces, vertices and edges.

Type of polyhedron

Edge length a

Result

Volume of the Cube (hexahedron)

Volume

Surface area

Circumradius

3.464102

Inradius

Faces

Vertices

Edges

Calculations use the same unit as the edge length. Area is in square units and volume in cubic units.

How it works

A regular polyhedron has faces that are all congruent regular polygons with the same number of faces meeting at every vertex. Only five exist: tetrahedron, cube, octahedron, dodecahedron and icosahedron.
With edge length a, the volume scales with a cubed and the surface area with a squared, while the circumscribed and inscribed sphere radii scale with a.
The tetrahedron has volume a cubed divided by 6√2 and surface area √3 times a squared.
The cube has volume a cubed and surface area 6 times a squared.
The octahedron has volume √2 divided by 3 times a cubed and surface area 2√3 times a squared.
The dodecahedron and icosahedron formulas involve the golden ratio φ. The circumscribed sphere passes through every vertex and the inscribed sphere touches every face.
Results use the same unit as the edge length, so area is in square units and volume in cubic units.

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