In geometry, the gyroelongated pentagonal cupola is one of the Johnson solids (J<sub>24</sub>). As the name suggests, it can be constructed by gyroelongating a pentagonal cupola (J<sub>5</sub>) by attaching a decagonal antiprism to its base. It can also be seen as a gyroelongated pentagonal bicupola (J<sub>46</sub>) with one pentagonal cupola removed.
thumb|left|3D model of a gyroelongated pentagonal cupola
Area and volume
With edge length a, the surface area is
:<math>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,</math>
and the volume is
:<math>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.</math>
Dual polyhedron
The dual of the gyroelongated pentagonal cupola has 25 faces: 10 kites, 5 rhombi, and 10 pentagons.
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!Dual gyroelongated pentagonal cupola
!Net of dual
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