Augmented pentagonal prism explained

Type:	Johnson
Faces:	4 triangles 4 squares 2 pentagons
Edges:	19
Vertices:	11
Properties:	convex
Net:	Johnson solid 52 net.png

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.

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. 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. 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} .

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:$$\backslash fraca^2\; \backslash approx\; 9.173a^2.$$Its volume can be obtained by slicing it into a regular pentagonal prism and an equilateral square pyramid, and adding their volume subsequently:$$\backslash fraca^3\; \backslash approx\; 1.9562a^3.$$

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:

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