Elongated pentagonal gyrobicupola
Type	Johnson J38 –J39 –J40
Faces	10triangles 20squares 2pentagons
Edges	60
Vertices	30
Vertex configuration	20(3.43) 10(3.4.5.4)
Symmetry group	D5d
Dual polyhedron	-
Properties	convex
Net

Ingeometry, theelongated pentagonal gyrobicupola is one of theJohnson solids (J₃₉). As the name suggests, it can be constructed by elongating apentagonal gyrobicupola (J₃₁) by inserting adecagonal prism between its congruent halves. Rotating one of thepentagonal cupolae (J₅) through 36 degrees before inserting the prism yields anelongated pentagonal orthobicupola (J₃₈).

AJohnson solid is one of 92 strictlyconvexpolyhedra that are composed ofregular polygon faces but are notuniform polyhedra (that is, they are notPlatonic solids,Archimedean solids,prisms, orantiprisms). They were named byNorman Johnson, who first listed these polyhedra in 1966.^[1]

The followingformulae forvolume andsurface area can be used if allfaces areregular, with edge lengtha:^[2]

${\displaystyle V=\frac{1}{6}\left(10+8\sqrt{5}+15\sqrt{5+2\sqrt{5}}\right)a^3\approx \mathrm{12.3423...}{a}^3}$

${\displaystyle A=\left(20+\sqrt{\frac{5}{2}\left(10+\sqrt{5}+\sqrt{75+30\sqrt{5}}\right)}\right)a^2\approx \mathrm{27.7711...}{a}^2}$

• Weisstein, Eric W., "Elongated pentagonal gyrobicupola" ("Johnson solid") atMathWorld.

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