Elongated pentagonal pyramid

File:J9 elongated pentagonal pyramid.stl

3D model of an 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.Template:R 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.Template:R It remains convex, with the faces are all regular polygons, so the elongated pentagonal pyramid is Johnson solid, enumerated as the sixteenth Johnson solid $J_{16}$.Template:R

For edge length $\ell$, 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:Template:R $$\begin{array}{rl}A& =\frac{20+5\sqrt{3}+\sqrt{25+10\sqrt{5}}}{4}{\ell }^2\approx 8.886{\ell }^2,\\ V& =\frac{5+\sqrt{5}+6\sqrt{25+10\sqrt{5}}}{24}{\ell }^3\approx 2.022{\ell }^3.\end{array}$$

The elongated pentagonal pyramid has a dihedral between its adjacent faces:Template:R

• 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°.

External links

• Template:Mathworld2