gyroelongated square cupola

{{Infobox polyhedron

|image=gyroelongated_square_cupola.png

|type=Johnson
J₂₂ - J₂₃ - J₂₄

|faces=3x4+8 triangles
1+4 squares
1 octagon

|edges=44

|vertices=20

|symmetry=C_4v|

|vertex_config=4(3.4³)
2.4(3³.8)
8(3⁴.4)

|dual=-

|properties=convex

|net=Johnson solid 23 net.png

}}

In geometry, the gyroelongated square cupola is one of the Johnson solids (J₂₃). As the name suggests, it can be constructed by gyroelongating a square cupola (J₄) by attaching an octagonal antiprism to its base. It can also be seen as a gyroelongated square bicupola (J₄₅) with one square bicupola removed.

Area and Volume

The surface area is,

: $A=\left(7+2\sqrt{2}+5\sqrt{3}\right)a^2\approx 18.4886811...a^2.$

The volume is the sum of the volume of a square cupola and the volume of an octagonal prism,

: $V=\left(1+\frac{2}{3}\sqrt{2} + \frac{2}{3}\sqrt{4+2\sqrt{2}+2\sqrt{146+103\sqrt{2}}}\right)a^3\approx6.2107658...a^3.$

The dual of the gyroelongated square cupola has 20 faces: 8 kites, 4 rhombi, and 8 pentagons.

class=wikitable width=320

valign=top

!Dual gyroelongated square cupola

!Net of dual

valign=top

{{mathworld2 | urlname2 = JohnsonSolid | title2 = Johnson solid| urlname = GyroelongatedSquareCupola | title = Gyroelongated square cupola }}