elongated pentagonal pyramid

{{Short description|9th Johnson solid (11 faces)}}

{{Infobox polyhedron

|image=elongated_pentagonal_pyramid.png

|type=Johnson
{{math|elongated square pyramid – J{{sub|9}} – gyroelongated square pyramid}}

|faces=5 triangles
5 squares
1 pentagon

|edges=20

|vertices=11

|symmetry={{math|cyclic symmetries, [5], (*55)}}

|rotation_group={{math|C{{sub|5}}, [5]{{sup|+}}, (55)}}

|vertex_config={{math|5(4{{sup|2}}.5)
5(3{{sup|2}}.4{{sup|2}})
1(3{{sup|5}})}}

|dual=self

|properties=convex

|net=Elongated_Pentagonal_Pyramid_Net.svg

}}

File:J9 elongated pentagonal pyramid.stl

The elongated pentagonal pyramid is a polyhedron constructed by attaching one pentagonal pyramid onto one of the pentagonal prism's bases, a process known as elongation. It is an example of composite polyhedron.{{r|timofeenko-2010|rajwade}} This construction involves the removal of one pentagonal face and replacing it with the pyramid. The resulting polyhedron has five equilateral triangles, five squares, and one pentagon as its faces.{{r|berman}} It remains convex, with the faces are all regular polygons, so the elongated pentagonal pyramid is Johnson solid, enumerated as the sixteenth Johnson solid $J\_\{16\}$.{{r|uehara}}

For edge length $\backslash ell$, an elongated pentagonal pyramid has a surface area $A$ by summing the area of all faces, and volume $V$ by totaling the volume of a pentagonal pyramid's Johnson solid and regular pentagonal prism:{{r|berman}}

$$\backslash begin\{align\}$$

A &= \frac{20 + 5\sqrt{3} + \sqrt{25 + 10\sqrt{5}}}{4}\ell^2 \approx 8.886\ell^2, \\

V &= \frac{5 + \sqrt{5} + 6\sqrt{25 + 10\sqrt{5}}}{24}\ell^3 \approx 2.022\ell^3.

\end{align}

The elongated pentagonal pyramid has a dihedral between its adjacent faces:{{r|johnson}}

• the dihedral angle between adjacent squares is the internal angle of the prism's pentagonal base, 108°;
• the dihedral angle between the pentagon and a square is the right angle, 90°;
• the dihedral angle between adjacent triangles is that of a regular icosahedron, 138.19°; and
• the dihedral angle between a triangle and an adjacent square is the sum of the angle between those in a pentagonal pyramid and the angle between the base of and the lateral face of a prism, 127.37°.