Pentagonal Prism Volume Calculator

This pentagonal prism volume calculator works out the volume of a prism with a regular pentagonal cross section. Enter the pentagon side length and the length of the prism, choose your unit, and it returns the volume together with the base area. A prism keeps the same cross section along its whole length, so the volume is the area of the pentagonal base times the length. For a regular pentagon the base area is one quarter times five times the side squared, times the cotangent of pi over five, and the calculator evaluates that for you. It is unit agnostic, so metres give cubic metres and centimetres give cubic centimetres. Pentagonal prisms appear in architecture, packaging, gemstone cuts and some structural sections, so this is useful for design and engineering as well as geometry classes. The side length is one edge of the pentagon, and the height is the length of the prism along its axis. The formula, a worked example and the assumptions are set out below so you can check the number and understand it.

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Base area 
Prism height10 m

The formula

The volume of a prism is the base area times the height. For a regular pentagon of side a the base area is (1/4)(5) a² cot(pi/5), so V = (1/4)(5) a² cot(pi/5) h. The base area constant is about 1.720477 times the side squared.

Worked example

A pentagonal prism with a side of 6 has a base area of about 61.94. Over a length of 10 the volume is about 619.37. Enter 6 and 10 above to confirm.

Frequently asked questions

What is the pentagonal prism volume formula?

V = the regular pentagon base area times the prism length. The base area is one quarter times five times the side squared, times the cotangent of pi over five.

Does it assume a regular pentagon?

Yes. It assumes a regular pentagon with five equal sides. An irregular cross section needs its own base area, then multiplied by the length.

What is the base area constant?

About 1.720477 times the side squared for a regular pentagon.

Who this calculator is for

This calculator is for students, architects and designers who need the volume of a regular pentagonal prism.

What this calculator assumes

  • The cross section is a regular pentagon with five equal sides.
  • You enter the pentagon side length and the prism length.
  • The volume is in the cube of your chosen unit.
  • Results are rounded for display.

Formula and sources

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