Pentagonal prism

If all faces are correct, the pentagonal prism becomes a semi-regular polyhedron . More generally, a prism is a , the third in the list of infinite prisms formed by square sides and two regular polygons as the bases of the prism. The pentagonal prism can be considered as a truncated pentagonal osohedron , represented by the Schleafly symbol t {2,5}. Alternatively, this prism can be considered as a direct product of a regular pentagon of a segment with the Schleafly symbol {5} x {}. The dual polyhedron of a pentagonal prism is a pentagonal bipyramid .

The symmetry group of the direct pentagonal prism is D _{5h of} order 20. The rotation group is D _{5 of} order 10.

The volume, as for all prisms, is equal to the product of the area of ​​the pentagonal base by the height (or the length of the edge perpendicular to the base). For a uniform pentagonal prism with edges of length h, the volume formula

${\displaystyle \frac{{\mathrm{<span\; class="mwe-math-element"><span\; class="mwe-math-mathml-inline\; mwe-math-mathml-a11y"\; style="display:\; none;">h</span></span>}}^{\mathrm{<span\; class="mwe-math-element"><span\; class="mwe-math-mathml-inline\; mwe-math-mathml-a11y"\; style="display:\; none;">3</span></span>}}}{\mathrm{<span\; class="mwe-math-element"><span\; class="mwe-math-mathml-inline\; mwe-math-mathml-a11y"\; style="display:\; none;">four</span></span>}}\sqrt{\mathrm{<span\; class="mwe-math-element"><span\; class="mwe-math-mathml-inline\; mwe-math-mathml-a11y"\; style="display:\; none;">five</span></span>}<span\; class="mwe-math-element"><span\; class="mwe-math-mathml-inline\; mwe-math-mathml-a11y"\; style="display:\; none;">(</span></span>\mathrm{<span\; class="mwe-math-element"><span\; class="mwe-math-mathml-inline\; mwe-math-mathml-a11y"\; style="display:\; none;">five</span></span>}<span\; class="mwe-math-element"><span\; class="mwe-math-mathml-inline\; mwe-math-mathml-a11y"\; style="display:\; none;">+</span></span>\mathrm{<span\; class="mwe-math-element"><span\; class="mwe-math-mathml-inline\; mwe-math-mathml-a11y"\; style="display:\; none;">2</span></span>}\sqrt{\mathrm{<span\; class="mwe-math-element"><span\; class="mwe-math-mathml-inline\; mwe-math-mathml-a11y"\; style="display:\; none;">five</span></span>}}<span\; class="mwe-math-element"><span\; class="mwe-math-mathml-inline\; mwe-math-mathml-a11y"\; style="display:\; none;">)</span></span>}}${\ displaystyle {\ frac {h ^ {3}} {4}} {\ sqrt {5 (5 + 2 {\ sqrt {5}})}}}  

Inhomogeneous pentagonal prisms are called pentaprisms and are used in optics to rotate an image at a right angle without changing chirality .

In 4-dimensional polyhedra

A pentagonal prism is found as a cell of four non-prismatic in four-dimensional space:

 

The toroidal polyhedron has pentagonal dihedral symmetry and has the same vertices as a homogeneous pentagonal prism .

• Weisstein, Eric W. Pentagonal prism on the Wolfram MathWorld website.
• Pentagonal Prism Polyhedron Model - works in your web browser