Gyroelongated pentagonal pyramid



In geometry, the gyroelongated pentagonal pyramid is a polyhedron constructed by taking a pentagonal pyramid and attaching it into pentagonal antiprism to its base. An alternative name is diminished icosahedron because it can be constructed by removing a pentagonal pyramid from a regular icosahedron.

Construction
The gyroelongated pentagonal pyramid can be constructed from a pentagonal antiprism by attaching a pentagonal pyramid onto its pentagonal face. This pyramid covers the pentagonal faces, so the resulting polyhedron has 15 equilateral triangles and 1 regular pentagon as its faces. Another way to construct it is started from the regular icosahedron by cutting off one of two pentagonal pyramids, a process known as diminishment; for this reason, it is also called the diminished icosahedron. Because the resulting polyhedron has the property of convexity and its faces are regular polygons, the gyroelongated pentagonal pyramid is a Johnson solid, enumerated as the 11th Johnson solid $$ J_{11} $$.

Properties
The surface area of a gyroelongated pentagonal pyramid $$ A $$ can be obtained by summing the area of 15 equilateral triangles and 1 regular pentagon. Its volume $$ V $$ can be ascertained either by slicing it off into both a pentagonal antiprism and a pentagonal pyramid, after which adding them up; or by subtracting the volume of a regular icosahedron to a pentagonal pyramid. With edge length $$ a $$, they are: $$ \begin{align} A &= \frac{15 \sqrt{3} + \sqrt{5(5 + 2\sqrt{5})}}{4}a^2 \approx 8.215a^2, \\ V &= \frac{25 + 9\sqrt{5}}{24}a^3 \approx 1.880a^3. \end{align} $$

It has the same three-dimensional symmetry group as the pentagonal pyramid: the cyclic group $$ C_{5 \mathrm{v}} $$ of order 10. Its dihedral angle can be obtained by involving the angle of a pentagonal antiprism and pentagonal pyramid: its dihedral angle between triangle-to-pentagon is the pentagonal antiprism's angle between that 100.8°, and its dihedral angle between triangle-to-triangle is the pentagonal pyramid's angle 138.2°.