Biaugmented pentagonal prism

The biaugmented pentagonal prism can be constructed from a pentagonal prism by attaching two equilateral square pyramids to each of its square faces, a process known as augmentation.^[1] These square pyramids cover the square face of the prism, so the resulting polyhedron has eight equilateral triangles, three squares, and two regular pentagons as its faces.^[2] A convex polyhedron in which all faces are regular is Johnson solid, and the augmented pentagonal prism is among them, enumerated as 53rd Johnson solid ${\displaystyle J_{53}}$.^[3]

An biaugmented pentagonal prism with edge length ${\displaystyle a}$ has a surface area, calculated by adding the area of four equilateral triangles, four squares, and two regular pentagons:^[2] $${\displaystyle {\frac {6+4{\sqrt {3}}+{\sqrt {5+2{\sqrt {5}}}}}{2}}a^{2}\approx 9.9051a^{2}.}$$ Its volume can be obtained by slicing it into a regular pentagonal prism and an equilateral square pyramid, and adding their volume subsequently:^[2] $${\displaystyle {\frac {\sqrt {257+90{\sqrt {5}}+24{\sqrt {50+20{\sqrt {5}}}}}}{12}}a^{3}\approx 2.1919a^{3}.}$$

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:^[4]

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