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 elongated 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 (J₃₉). As the name suggests, it can be constructed by elongating a pentagonal gyrobicupola

Pentagonal gyrobicupola

In geometry, the pentagonal gyrobicupola is one of the Johnson solids . Like the pentagonal orthobicupola , it can be obtained by joining two pentagonal cupolae along their bases...

 (J₃₁) by inserting a decagonal prism

Decagonal prism

In geometry, the decagonal prism is the eighth in an infinite set of prisms, formed by ten square side faces and two regular decagon caps. With twelve faces, it is one of many nonregular dodecahedra.If faces are all regular, it is a semiregular polyhedron....

 between its congruent halves. Rotating one of the 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) through 36 degrees before inserting the prism yields an elongated pentagonal orthobicupola

Elongated pentagonal orthobicupola

In geometry, the elongated pentagonal orthobicupola is one of the Johnson solids . As the name suggests, it can be constructed by elongating a pentagonal orthobicupola by inserting a decagonal prism between its two congruent halves...

  (J₃₈).

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 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: