Pentagonal Prism Volume Calculator

Pentagonal Prism: how to find the surface area and volume when one side length is given, and how to find the surface area and side length when the volume is given.

Pentagonal Prism Calculation Method

Pentagonal Prism: calculate the volume and surface area from the side length.

Formula for the volume

Pentagonal Prismformula for the area

The side length (a) of Pentagonal Prism is 0 .

The volume (V) of Pentagonal Prism is 0 .

The area (S) of Pentagonal Prism is 0 .

The height (h) of Pentagonal Prism is 0 .

About This Calculator

A regular pentagonal prism is a solid with two congruent regular pentagons as bases, joined by five rectangles. Because every base edge has the same length, the volume follows directly from that one value and the prism length.

Enter the pentagon side and the prism length to get the volume and the total surface area. The area of a regular pentagon is (5/4)·cot 36° ≈ 1.7205 times the side squared, which is where the constant in the formula comes from.

A regular pentagonal prism uses the base area (1/4)√(5(5 + 2√5))a² ≈ 1.7205a², so a = 6 and h = 10 give a base of 61.94, a volume of 619.37 and a total surface area of 423.87. A regular pentagon does not tile the plane, which is why pentagonal prisms belong to decoration and architecture rather than to packing.

Key Formula:V = (5/4)·cot 36° · a² · h ≈ 1.7205 · a² · h · S = 2·A_base + 5·a·h

Frequently Asked Questions

How do I find the area of a regular pentagon?

A = (1/4)·√(5·(5+2√5))·a² ≈ 1.7205·a². With a 6 cm side that is about 61.94 cm². The same value comes from splitting the pentagon into five isosceles triangles with a 72° apex angle.

Where do pentagonal prisms appear in practice?

In pentaprisms inside DSLR viewfinders, which flip the image upright, in decorative mouldings and in extruded aluminium profiles. In nature, five-fold symmetry is rarer than six-fold because regular pentagons cannot tile a flat surface without gaps.

Why is the pentagon coefficient about 1.7205?

Splitting the pentagon into five isosceles triangles from the centre gives (5/4)a² cot 36°, and cot 36° ≈ 1.3764, so the coefficient is 1.25 × 1.3764 ≈ 1.7205.

How does a pentagonal prism differ from a pentagonal pyramid?

The prism has two congruent pentagonal faces joined by five rectangles, while the pyramid has one pentagonal base and five triangles meeting at an apex. For the same base and height the pyramid holds exactly one third of the volume.