In general, an icosahedron is a 20-faced polyhedron (where icos- derives from the Greek word for "twenty" and -hedron comes from the Indo-European word for "seat"). Examples illustrated above include the decagonal dipyramid, elongated triangular gyrobicupola (Johnson solid ), elongated triangular orthobicupola (), gyroelongated triangular cupola (), Jessen's orthogonal icosahedron, metabiaugmented dodecahedron (), nonagonal antiprism, parabiaugmented dodecahedron (), 18-gonal prism, 19-gonal pyramid, regular icosahedron, and rhombic icosahedron.

"The" icosahedron (more properly called the regular icosahedron) is the regular polyhedron and Platonic solid illustrated above having 12 polyhedron vertices, 30 polyhedron edges, and 20 equivalent equilateral triangle faces, .

The regular icosahedron is also uniform polyhedron and Wenninger model . It is described by the Schläfli symbol and Wythoff symbol . Coxeter et al. (1999) have shown that there are 58 icosahedron stellations (giving a total of 59 solids when the icosahedron itself is included).

Two icosahedra constructed in origami are illustrated above (Gurkewitz and Arnstein 1995, p. 53). This construction uses 30 triangle edge modules, each made from a single sheet of origami paper.

Two icosahedra appears as polyhedral "stars" in M. C. Escher's 1948 wood engraving "Stars" (Forty 2003, Plate 43).

There are 43380 distinct nets for the icosahedron, the same number as for the dodecahedron (Bouzette and Vandamme, Hippenmeyer 1979, Buekenhout and Parker 1998).

The icosahedron has the icosahedral group of symmetries. The connectivity of the vertices is given by the icosahedral graph.

The dual polyhedron of an icosahedron with unit edge lengths is the dodecahedron with edge lengths , where is the golden ratio. As a result, the centers of the faces of an icosahedron form a dodecahedron, and vice versa, illustrated above (Steinhaus 1999, pp. 199-201).

In particular, fifteen golden rectangles span the interior of the icosahedron. These rectangles have 30 edges, and each edge pairs up with its opposite edge to form a golden rectangle

There are 59 distinct icosahedra when each triangle is colored differently (Coxeter 1969). More general question of polyhedron coloring can be addressed using the Pólya enumeration theorem.

Taken eight at a time, the centers of the faces of an icosahedron comprise the vertices of a cube. This leads to the beautiful cube 5-compound and is the basis for Jessen's orthogonal icosahedron.

A plane perpendicular to a axis of an icosahedron cuts the solid in a regular decagonal cross section (Holden 1991, pp. 24-25).

The long diagonals of the faces of the rhombic triacontahedron give the edges of an icosahedron (Steinhaus 1999, pp. 209-210).

The following table gives polyhedra which can be constructed by cumulation of an icosahedron by pyramids of given heights .

		result
		great dodecahedron
		small triambic icosahedron
		60-faced star deltahedron
	3	great stellated dodecahedron

A construction for an icosahedron with side length places the end vertices at ) and the central vertices around two staggered circles of radii and heights . By a suitable rotation, the polyhedron vertices of an icosahedron of side length 2 can also be placed at , , and , where is the golden ratio. These points divide the polyhedron edges of an octahedron into segments with lengths in the ratio . Another orientation of the icosahedron places two opposite triangular faces in an orientation parallel to the -plane. In this orientation, the distance from the top plane to the triangle of vertices below it is , equal to the circumradius of a face. The circumradius of is given by

(1)

To derive the volume of an icosahedron having edge length , consider the orientation so that two polyhedron vertices are oriented on top and bottom. The vertical distance between the top and bottom pentagonal dipyramids is then given by

(2)

where

(3)

is the height of an equilateral triangle, and the sagitta of the pentagon is

(4)

giving

(5)

Plugging (3) and (5) into (2) gives

(6)

which is identical to the radius of a pentagon of side . The circumradius is then

(7)

where

(8)

is the height of a pentagonal dipyramid. Therefore,

			(9)
			(10)

Taking the square root gives the circumradius

			(11)
			(12)
			(13)

The inradius is

(14)

The square of the midradius is

(15)

so

(16)

The dihedral angle is

(17)

The area of one face is the area of an equilateral triangle

(18)

The volume can be computed by taking 20 pyramids of height

(19)

Apollonius showed that for an icosahedron and a dodecahedron with the same inradius,

(20)

where is the volume and the surface area, with the actual ratio being

(21)

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Weisstein, Eric W. "Icosahedron." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/Icosahedron.html