Biaugmented pentagonal prism

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In geometry, the biaugmented pentagonal prism is a polyhedron constructed from a pentagonal prism by attaching two equilateral square pyramids onto each of its square faces. It is an example of Johnson solid.

Construction

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.Template:R 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.Template:R 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 <math> J_{53} </math>.Template:R

Properties

An biaugmented pentagonal prism with edge length <math> a </math> has a surface area, calculated by adding the area of four equilateral triangles, four squares, and two regular pentagons:Template:R <math display="block"> \frac{6 + 4\sqrt{3} + \sqrt{5 + 2\sqrt{5}}}{2}a^2 \approx 9.9051a^2. </math> 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 <math display="block"> \frac{\sqrt{257 + 90\sqrt{5} + 24\sqrt{50 + 20\sqrt{5}}}}{12}a^3 \approx 2.1919a^3. </math>

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

External links

• {{#invoke:Template wrapper|{{#if:|list|wrap}}|_template=cite web

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• • {{#invoke:Template wrapper|{{#if:|list|wrap}}|_template=cite web

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