Three-dimensional geometric shapes

Platonic solids

Keywords: Platonic solids, also called the regular solids or regular polyhedra

Name Tetrahedron Hexahedron
(Cube)
Octahedron Dodecahedron Icosahedron
Picture Tetrahedron Hexahedron (Cube) Octahedron Dodecahedron Icosahedron
Net Tetrahedron net Hexahedron (Cube) net Octahedron net Dodecahedron net Icosahedron net
Faces 4 6 8 12 20
Face type
equilateral triangle square equilateral triangle equilateral pentagon equilateral triangle
Duality tetrahedron octahedron hexahedron icosahedron dodecahedron
Edges 6 12 12 30 30
Vertices 4 8 6 20 12
Edges from vertex 3 3 4 3 5
Number of diagonals 0 4 3 100 36
Dihedral angle ≈70°31'43,61'' 90° ≈109°28'16,39'' ≈116°33'55,84'' ≈138°11'22,87''
Surface area 3 a 2 6 a 2 2 3 a 2 3 25 + 10 5 a 2 5 3 a 2
Volume 2 a 3 12 a 3 2 a 3 3 ( 15 + 7 5 ) a 3 4 ( 15 + 5 5 ) a 3 12
Circumradiu 6 a 4 3 a 2 3 a 2 3 ( 1 + 5 ) a 4 10 + 2 5 a 4
Inradiu 6 a 12 a 2 6 a 6 1 2 25 + 11 5 10 a 42 + 18 5 12 a
Midradius 2 a 4 2 a 2 a 2 ( 5 + 3 ) a 4 ( 1 + 5 ) a 4