Decahedra

A decahedron is a polyhedron with ten faces. Like other polyhedra, decahedra can be convex or concave, and their faces may be triangles, quadrilaterals, or other polygons. The term describes only the number of faces, not a single unique shape, so there is more than one possible decahedron.

Two notable convex decahedra are the pentagonal dipyramid and the pentagonal trapezohedron. The pentagonal dipyramid consists of two pentagonal pyramids joined base-to-base, giving ten triangular faces and a high degree of symmetry. The pentagonal trapezohedron has ten congruent kite-shaped faces and is isohedral, meaning all faces are equivalent under its symmetry group; it is the standard shape used for the ten-sided die (d10) in many tabletop games.

In general, decahedra exhibit a range of topologies and face configurations. Some decahedra are deltahedra, composed entirely of triangular faces (such as the pentagonal dipyramid), while others mix triangles with quadrilaterals or kite faces, leading to a variety of possible shapes and symmetry properties. There is no single regular decahedron, since among the Platonic solids the regular ones do not include a ten-faced form.

Decahedra appear in mathematical discussions of polyhedra as well as in practical contexts, notably in dice design and decorative geometry. They illustrate how a fixed count of faces allows many distinct spatial realizations with different symmetries and face types.

See also: Polyhedron, Deltahedron, Trapezohedron, Dodecahedron, D10 die.