Augmented pentagonal prism

In geometry, the augmented pentagonal prism is a polyhedron that can be constructed by attaching an equilateral square pyramid onto the square face of pentagonal prism. It is an example of Johnson solid.

File:J52 augmented pentagonal prism.stl

3D model of an augmented pentagonal prism

Construction

The augmented pentagonal prism can be constructed from a pentagonal prism by attaching an equilateral square pyramid to one of its square faces, a process known as augmentation.Template:R This square pyramid covers the square face of the prism, so the resulting polyhedron has four equilateral triangles, four squares, and two regular pentagons as its faces.Template:R A convex polyhedron in which all faces are regular is Johnson solid, and the augmented pentagonal prism is among them, enumerated as 52nd Johnson solid $J_{52}$.Template:R

Properties

An augmented pentagonal prism with edge length $a$ has a surface area, calculated by adding the area of four equilateral triangles, four squares, and two regular pentagons:Template:R$$\frac{8+2\sqrt{3}+\sqrt{5+2\sqrt{5}}}{2}{a}^2\approx 9.173a^2.$$ Its volume can be obtained by slicing it into a regular pentagonal prism and an equilateral square pyramid, and adding their volume subsequently:Template:R$$\frac{\sqrt{233+90\sqrt{5}+12\sqrt{50+20\sqrt{5}}}}{12}{a}^3\approx 1.9562a^3.$$

The dihedral angle of an augmented pentagonal prism can be calculated by adding the dihedral angle of an equilateral square pyramid and the regular pentagonal prism:Template:R

• the dihedral angle of an augmented pentagonal prism between two adjacent triangular faces is that of an equilateral square pyramid between two adjacent triangular faces, $\arccos (-\frac{1}{3})\approx 109.5^{\circ }$,
• the dihedral angle of an augmented pentagonal prism between two adjacent square faces is the internal angle of a regular pentagon $\frac{3\pi }{5}=108^{\circ }$.
• the dihedral angle of an augmented pentagonal prism between square-to-pentagon is that of a regular pentagonal prism between its base and its lateral faces $\frac{\pi }{2}=90^{\circ }$.
• the dihedral angle of an augmented pentagonal prism between pentagon-to-triangle is $\arctan \left(\sqrt{2}\right)+\frac{\pi }{2}\approx 144.7^{\circ }$, for which adding the dihedral angle of an equilateral square pyramid between its base and its lateral face $\arctan \left(\sqrt{2}\right)\approx 54.7^{\circ }$, and the dihedral angle of a regular pentagonal prism between its base and its lateral face.
• the dihedral angle of an augmented pentagonal prism between square-to-triangle is $\arctan \left(\sqrt{2}\right)+\frac{3\pi }{5}\approx 162.7^{\circ }$, for which adding the dihedral angle of an equilateral square pyramid between its base and its lateral face, and the dihedral angle of a regular pentagonal prism between two adjacent squares.