Elongated pentagonal pyramid

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.^[2][3] 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.^[4] It remains convex, with the faces are all regular polygons, so the elongated pentagonal pyramid is Johnson solid, enumerated as the ninth Johnson solid ${\displaystyle J_{9}}$.^[5]

For edge length ${\displaystyle \ell }$, an elongated pentagonal pyramid has a surface area ${\displaystyle A}$ by summing the area of all faces, and volume ${\displaystyle V}$ by totaling the volume of a pentagonal pyramid's Johnson solid and regular pentagonal prism:^[4] $${\displaystyle {\begin{aligned}A&={\frac {20+5{\sqrt {3}}+{\sqrt {25+10{\sqrt {5}}}}}{4}}\ell ^{2}\approx 8.886\ell ^{2},\\V&={\frac {5+{\sqrt {5}}+6{\sqrt {25+10{\sqrt {5}}}}}{24}}\ell ^{3}\approx 2.022\ell ^{3}.\end{aligned}}}$$

The elongated pentagonal pyramid has a dihedral between its adjacent faces:^[6]

• 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°.