Deltahedron in the context of Triaugmented triangular prism

The regular icosahedron (or simply icosahedron) is a convex polyhedron that can be constructed from pentagonal antiprism by attaching two pentagonal pyramids with regular faces to each of its pentagonal faces, or by putting points onto the cube. The resulting polyhedron has 20 equilateral triangles as its faces, 30 edges, and 12 vertices. It is an example of a Platonic solid and of a deltahedron. The icosahedral graph represents the skeleton of a regular icosahedron.

Many polyhedra and other related figures are constructed from the regular icosahedron, including its 59 stellations. The great dodecahedron, one of the Kepler–Poinsot polyhedra, is constructed by either stellation of the regular dodecahedron or faceting of the icosahedron. Some of the Johnson solids can be constructed by removing the pentagonal pyramids. The regular icosahedron's dual polyhedron is the regular dodecahedron, and their relation has a historical background in the comparison mensuration. It is analogous to a four-dimensional polytope, the 600-cell.

An equilateral triangle is a triangle in which all three sides have the same length, and all three angles are equal. Because of these properties, the equilateral triangle is a regular polygon, occasionally known as the regular triangle. It is the special case of an isosceles triangle by modern definition, creating more special properties.

The equilateral triangle can be found in various tilings, and in polyhedrons such as the deltahedron and antiprism. It appears in real life in popular culture, architecture, and the study of stereochemistry resembling the molecular known as the trigonal planar molecular geometry.