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 orthocupolarotunda 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 (J₃₂). As the name suggests, it can be constructed by joining a pentagonal cupola

Pentagonal cupola

In geometry, the pentagonal cupola is one of the Johnson solids . It can be obtained as a slice of the rhombicosidodecahedron.The 92 Johnson solids were named and described by Norman Johnson in 1966....

 (J₅) and a pentagonal rotunda

Pentagonal rotunda

In geometry, the pentagonal rotunda is one of the Johnson solids . It can be seen as half an icosidodecahedron.The 92 Johnson solids were named and described by Norman Johnson in 1966.-Formulae:...

 (J₆) along their 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°...

al bases, matching the pentagonal faces. A 36-degree rotation of one of the halves before the joining yields a pentagonal gyrocupolarotunda

Pentagonal gyrocupolarotunda

In geometry, the pentagonal gyrocupolarotunda is one of the Johnson solids . Like the pentagonal orthocupolarotunda , it can be constructed by joining a pentagonal cupola and a pentagonal rotunda along their decagonal bases...

 (J₃₃).

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....

 and 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...

 can be used if all faces 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: