In geometry

Geometry

Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers ....

, the pentagonal cupola

Cupola (geometry)

In geometry, a cupola is a solid formed by joining two polygons, one with twice as many edges as the other, by an alternating band of triangles and rectangles...

is one of the Johnson solid

Johnson solid

In geometry, a Johnson solid is a strictly convex polyhedron, each face of which is a regular polygon, but which is not uniform, i.e., not a Platonic solid, Archimedean solid, prism or antiprism. There is no requirement that each face must be the same polygon, or that the same polygons join around...

s (J₅). It can be obtained as a slice of the rhombicosidodecahedron

Rhombicosidodecahedron

In geometry, the rhombicosidodecahedron, or small rhombicosidodecahedron, is an Archimedean solid, one of thirteen convex isogonal nonprismatic solids constructed of two or more types of regular polygon faces....

.

The 92 Johnson solids were named and described by Norman Johnson in 1966.

The pentagonal cupola consists of 5 equilateral triangles, 5 square

Square (geometry)

In geometry, a square is a regular quadrilateral. This means that it has four equal sides and four equal angles...

s, 1 pentagon

Pentagon

In geometry, a pentagon is any five-sided polygon. A pentagon may be simple or self-intersecting. The sum of the internal angles in a simple pentagon is 540°. A pentagram is an example of a self-intersecting pentagon.- Regular pentagons :In a regular pentagon, all sides are equal in length and...

, and 1 decagon

Decagon

In geometry, a decagon is any polygon with ten sides and ten angles, and usually refers to a regular decagon, having all sides of equal length and each internal angle equal to 144°...

.

Formulae

The following formula

Formula

In mathematics, a formula is an entity constructed using the symbols and formation rules of a given logical language....

e for volume

Volume

Volume is the quantity of three-dimensional space enclosed by some closed boundary, for example, the space that a substance or shape occupies or contains....

, surface area

Surface area

Surface area is the measure of how much exposed area a solid object has, expressed in square units. Mathematical description of the surface area is considerably more involved than the definition of arc length of a curve. For polyhedra the surface area is the sum of the areas of its faces...

 and circumradius

Circumscribed sphere

In geometry, a circumscribed sphere of a polyhedron is a sphere that contains the polyhedron and touches each of the polyhedron's vertices. The word circumsphere is sometimes used to mean the same thing...

 can be used if all faces

Face (geometry)

In geometry, a face of a polyhedron is any of the polygons that make up its boundaries. For example, any of the squares that bound a cube is a face of the cube...

 are regular

Regular polygon

A regular polygon is a polygon that is equiangular and equilateral . Regular polygons may be convex or star.-General properties:...

, with edge length a:

Dual polyhedron

The dual of the pentagonal cupola has 20 triangular faces:

Dual pentagonal cupola	Net of dual