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  2. Regular icosahedron - Wikipedia

    en.wikipedia.org/wiki/Regular_icosahedron

    Therefore, the surface area () of a regular icosahedron is 20 times that of each of its equilateral triangle faces. The volume ( V ) {\displaystyle (V)} of a regular icosahedron can be obtained as 20 times that of a pyramid whose base is one of its faces and whose apex is the icosahedron's center; or as the sum of two uniform pentagonal ...

  3. Icosahedron - Wikipedia

    en.wikipedia.org/wiki/Icosahedron

    Three interlocking golden rectangles inscribed in a con­vex regular icosahedron. The convex regular icosahedron is usually referred to simply as the regular icosahedron, one of the five regular Platonic solids, and is represented by its Schläfli symbol {3, 5}, containing 20 triangular faces, with 5 faces meeting around each vertex.

  4. Surface area - Wikipedia

    en.wikipedia.org/wiki/Surface_area

    A sphere of radius r has surface area 4πr 2.. The surface area (symbol A) of a solid object is a measure of the total area that the surface of the object occupies. [1] The mathematical definition of surface area in the presence of curved surfaces is considerably more involved than the definition of arc length of one-dimensional curves, or of the surface area for polyhedra (i.e., objects with ...

  5. Truncated icosahedron - Wikipedia

    en.wikipedia.org/wiki/Truncated_icosahedron

    The surface area and the volume of the truncated icosahedron of edge length are: [2] = (+ +) = +. The sphericity of a polyhedron describes how closely a polyhedron resembles a sphere. It can be defined as the ratio of the surface area of a sphere with the same volume to the polyhedron's surface area, from which the value is between 0 and 1.

  6. Truncated icosidodecahedron - Wikipedia

    en.wikipedia.org/wiki/Truncated_icosidodecahedron

    In geometry, a truncated icosidodecahedron, rhombitruncated icosidodecahedron, [1] great rhombicosidodecahedron, [2] [3] omnitruncated dodecahedron or omnitruncated icosahedron [4] is an Archimedean solid, one of thirteen convex, isogonal, non-prismatic solids constructed by two or more types of regular polygon faces.

  7. Icosidodecahedron - Wikipedia

    en.wikipedia.org/wiki/Icosidodecahedron

    The icosidodecahedron is an Archimedean solid, meaning it is a highly symmetric and semi-regular polyhedron, and two or more different regular polygonal faces meet in a vertex. [5] The polygonal faces that meet for every vertex are two equilateral triangles and two regular pentagons, and the vertex figure of an icosidodecahedron is {{nowrap|(3 ...

  8. Final stellation of the icosahedron - Wikipedia

    en.wikipedia.org/wiki/Final_stellation_of_the...

    As a simple, visible surface polyhedron, the outward form of the final stellation is composed of 180 triangular faces, which are the outermost triangular regions in the stellation diagram. These join along 270 edges, which in turn meet at 92 vertices, with an Euler characteristic of 2.

  9. Platonic solid - Wikipedia

    en.wikipedia.org/wiki/Platonic_solid

    This follows from the spherical excess formula for a spherical polygon and the fact that the vertex figure of the polyhedron {p,q} is a regular q-gon. The solid angle of a face subtended from the center of a platonic solid is equal to the solid angle of a full sphere (4 π steradians) divided by the number of faces.

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