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Johnson solid

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Bilunabirotunda

In geometry, the bilunabirotunda is one of the Johnson solids (J_91). It is one of the elementary Johnson solids, which do not arise from "cut and paste" manipulations of the Platonic and Archimedean solids. However, it does have a strong relationship to the icosidodecahedron, an Archimedean solid. Either one of the two clusters of two pentagons and two triangles can be aligned with a congruent patch of faces on the icosidodecahedron.

Triangular hebesphenorotunda

In geometry, the triangular hebesphenorotunda is one of the Johnson solids (J_92). It is one of the elementary Johnson solids, which do not arise from "cut and paste" manipulations of the Platonic and Archimedean solids. However, it does have a strong relationship to the icosidodecahedron, an Archimedean solid. Most evident is the cluster of three pentagons and four triangles on one side of the solid. If these faces are aligned with a congruent patch of faces on the icosidodecahedron, then the hexagonal face will lie in the plane midway between two opposing triangular faces of the icosidodecahedron.

Elongated triangular pyramid

In geometry, the elongated triangular pyramid is one of the Johnson solids (J_7). As the name suggests, it can be constructed by elongating a tetrahedron by attaching a triangular prism to its base. Like any elongated pyramid, the resulting solid is topologically (but not geometrically) self-dual. The following formulae for volume and surface area can be used if all faces are regular, with edge length a: The height is given by If the edges are not the same length, use the individual formulae for the tetrahedron and triangular prism separately, and add the results together.

Elongated triangular bipyramid

In geometry, the elongated triangular bipyramid (or dipyramid) or triakis triangular prism is one of the Johnson solids (J_14), convex polyhedra whose faces are regular polygons. As the name suggests, it can be constructed by elongating a triangular bipyramid (J_12) by inserting a triangular prism between its congruent halves. The nirrosula, an African musical instrument woven out of strips of plant leaves, is made in the form of a series of elongated bipyramids with non-equilateral triangles as the faces of their end caps.

Elongated pentagonal cupola

In geometry, the elongated pentagonal cupola is one of the Johnson solids (J_20). As the name suggests, it can be constructed by elongating a pentagonal cupola (J_5) by attaching a decagonal prism to its base. The solid can also be seen as an elongated pentagonal orthobicupola (J_38) with its "lid" (another pentagonal cupola) removed. The following formulas for the volume and surface area can be used if all faces are regular, with edge length a: The dual of the elongated pentagonal cupola has 25 faces: 10 isosceles triangles, 5 kites, and 10 quadrilaterals.

Gyroelongated pentagonal cupola

In geometry, the gyroelongated pentagonal cupola is one of the Johnson solids (J24). As the name suggests, it can be constructed by gyroelongating a pentagonal cupola (J5) by attaching a decagonal antiprism to its base. It can also be seen as a gyroelongated pentagonal bicupola (J46) with one pentagonal cupola removed. With edge length a, the surface area is and the volume is The dual of the gyroelongated pentagonal cupola has 25 faces: 10 kites, 5 rhombi, and 10 pentagons.

Elongated pentagonal gyrobicupola

In geometry, the elongated pentagonal gyrobicupola is one of the Johnson solids (J_39). As the name suggests, it can be constructed by elongating a pentagonal gyrobicupola (J_31) by inserting a decagonal prism between its congruent halves. Rotating one of the pentagonal cupolae (J_5) through 36 degrees before inserting the prism yields an elongated pentagonal orthobicupola (J_38).

Metabiaugmented hexagonal prism

In geometry, the metabiaugmented hexagonal prism is one of the Johnson solids (J_56). As the name suggests, it can be constructed by doubly augmenting a hexagonal prism by attaching square pyramids (J_1) to two of its nonadjacent, nonparallel equatorial faces. Attaching the pyramids to opposite equatorial faces yields a parabiaugmented hexagonal prism. (The solid obtained by attaching pyramids to adjacent equatorial faces is not convex, and thus not a Johnson solid.

Triaugmented hexagonal prism

In geometry, the triaugmented hexagonal prism is one of the Johnson solids (J_57). As the name suggests, it can be constructed by triply augmenting a hexagonal prism by attaching square pyramids (J_1) to three of its nonadjacent equatorial faces.

Paragyrate diminished rhombicosidodecahedron

In geometry, the paragyrate diminished rhombicosidodecahedron is one of the Johnson solids (J_77). It can be constructed as a rhombicosidodecahedron with one pentagonal cupola rotated through 36 degrees, and the opposing pentagonal cupola removed.