Gyroelongated pentagonal cupola

{{Infobox polyhedron

|image=gyroelongated_pentagonal_cupola.png

|type=Johnson
J₂₃ - J₂₄ - J₂₅

|edges=55

|vertices=25

|symmetry=C_5v|

|vertex_config=5(3.4.5.4)
2.5(3³.10)
10(3⁴.4)

|dual=-

|properties=convex

|net=Johnson solid 24 net.png

}}

In geometry, the gyroelongated pentagonal cupola is one of the Johnson solids (J₂₄). As the name suggests, it can be constructed by gyroelongating a pentagonal cupola (J₅) by attaching a decagonal antiprism to its base. It can also be seen as a gyroelongated pentagonal bicupola (J₄₆) with one pentagonal cupola removed.

Area and Volume

With edge length a, the surface area is

: $A=\frac{1}{4}\left( 20+25\sqrt{3}+\left(10+\sqrt{5}\right)\sqrt{5+2\sqrt{5}}\right)a^2\approx25.240003791...a^2,$

and the volume is

: $V=\left(\frac{5}{6}+\frac{2}{3}\sqrt{5} + \frac{5}{6}\sqrt{2\sqrt{650+290\sqrt{5}}-2\sqrt{5}-2}\right) a^3\approx 9.073333194...a^3.$

The dual of the gyroelongated pentagonal cupola has 25 faces: 10 kites, 5 rhombi, and 10 pentagons.

class=wikitable width=320

valign=top

!Dual gyroelongated pentagonal cupola

!Net of dual

valign=top

{{mathworld2 | urlname2 = JohnsonSolid | title2 = Johnson solid| urlname = GyroelongatedPentagonalCupola | title = Gyroelongated pentagonal cupola }}