Pentagonal prism volume calculator
The Pentagonal prism volume calculator computes pentagonal prism volume from the relation V = (1/4) sqrt(5(5+2sqrt5)) a^2 h. It takes 2 inputs (side, height) and returns the volume. These calculators suit students learning how a formula behaves, tradespeople and makers working out how much material a shape will take, and anyone who needs a length, area, volume or angle for a figure they have measured or sketched. Enter your values below and the result updates instantly, and you can share a permalink that pre-fills the exact calculation. Put every length into the same unit before you calculate, because mixed units corrupt the answer without any warning, and check that you are entering the dimension the field actually asks for (radius rather than diameter, perpendicular height rather than a slanted edge); your answer is only as precise as the roughest measurement behind it, and squaring or cubing widens that gap, so round the figure you report rather than copying out every digit on screen. For example, with side = 2, height = 5, the volume works out to 34.409548, and the worked example further down the page shows every step so you can follow the arithmetic and reproduce it by hand. The method is the standard form documented by NIST DLMF, and the figure above each result carries the date it was last verified. A result describes the idealised figure your inputs define, so it makes no allowance for rounded corners, cut waste or an uneven surface on a physical object, and the inputs have to stay within the range a figure permits (lengths greater than zero, and side or angle combinations that can actually close into the shape); where the answer rests on an irrational constant such as pi or on a root, the value shown is a rounded form of an exact one.
With Side = 2, Height = 5, the result is 34.409548.
Applies to: any numeric inputs. Method source: NIST DLMF, checked 2026-07-09.
The formula
V = (1/4) sqrt(5(5+2sqrt5)) a^2 h
Worked example
With Side = 2, Height = 5:
- V = pentagon area x h
- Volume = 34.409548
This worked example is one of the automated golden-value tests this calculator must pass before it can publish.
What this assumes
- Inputs are real numbers in the units shown.
- The result is the exact value of V = (1/4) sqrt(5(5+2sqrt5)) a^2 h; general information, not professional advice.
Frequently asked questions
What formula does this use?
V = (1/4) sqrt(5(5+2sqrt5)) a^2 h, the standard form documented by NIST DLMF.
Does the result ever change over time?
No. This is a pure formula with no external rate, so the same inputs always give the same result.
Official sources and verification
- Method: NIST DLMF, checked 2026-07-09.
Reviewed by the CalculatorHub team, edited by James Graham, 2026-07-09. See our methodology. General information, not professional advice.