# Icosahedron

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In geometry, an icosahedron (Greek: εικοσάεδρον, from eikosi twenty + hedron seat; pronounced /ˌaɪkɵsəˈhiːdrən/ or /aɪˌkɒsəˈhiːdrən/; plural: -drons, -dra /-drə/) is a regular polyhedron with 20 identical equilateral triangular faces, 30 edges and 12 vertices. It is one of the five Platonic solids.

It has five triangular faces meeting at each vertex. It can be represented by its vertex figure as 3.3.3.3.3 or 35, and also by Schläfli symbol {3,5}. It is the dual of the dodecahedron, which is represented by {5,3}, having three pentagonal faces around each vertex.

## Contents

### Dimensions

If the edge length of a regular icosahedron is a, the radius of a circumscribed sphere (one that touches the icosahedron at all vertices) is

and the radius of an inscribed sphere (tangent to each of the icosahedron's faces) is

while the midradius, which touches the middle of each edge, is

where $\varphi$ (also called τ) is the golden ratio.

### Area and volume

The surface area A and the volume V of a regular icosahedron of edge length a are: