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 gyrobicupola 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 (J₃₁). Like the pentagonal orthobicupola

Pentagonal orthobicupola

In geometry, the pentagonal orthobicupola is one of the Johnson solids . As the name suggests, it can be constructed by joining two pentagonal cupolae along their decagonal bases, matching like faces...

 (J₃₀), it can be obtained by joining two 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....

e (J₅) along their bases. The difference is that in this solid, the two halves are rotated 36 degrees with respect to one another.

The pentagonal gyrobicupola is the third in an infinite set of gyrobicupolae

Bicupola (geometry)

In geometry, a bicupola is a solid formed by connecting two cupolae on their bases.There are two classes of bicupola because each cupola half is bordered by alternating triangles and squares...

.

The pentagonal gyrobicupola is what you get when you take a 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....

, chop out the middle parabidiminished rhombicosidodecahedron

Parabidiminished rhombicosidodecahedron

In geometry, the parabidiminished rhombicosidodecahedron is one of theJohnson solids .It can be constructed as a rhombicosidodecahedron with two opposing pentagonal cupola removed.Related Johnson solids are...

 (J₈₀), and paste the two opposing cupolae back together.

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

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

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: