Pentagonal Prism Volume Calculator
From a pentagon side and a prism length, get the volume, the cross-section area, and the total surface area — the three numbers that describe any regular pentagonal prism.
Two inputs, three answers
Enter the pentagon side and the prism length and the calculator returns the volume, the cross-section area (one pentagonal end), and the total surface area all at once.
Keep units consistent
The side and length are unit-agnostic — your answers come back in the same unit (squared for the area, cubed for the volume), so don't mix centimetres with inches.
What is a pentagonal prism volume calculator?
Side and length in, full prism out
A pentagonal prism volume calculator turns two measurements — the side length of a regular pentagon and the length of the prism — into the numbers that describe a whole pentagonal prism: how much it holds (volume), the area of one pentagonal face (cross-section area), and the area of all its outer faces (surface area). A regular pentagonal prism has two identical regular-pentagon ends joined by five equal rectangles, so once you know the side and the length every measurement is fixed. That makes those two inputs all you need for nut-shaped bolts, pencils, prism-shaped containers, packaging, and any geometry homework where a pentagonal prism shows up.
Enter the pentagon side and the prism length in any length unit to get the volume, cross-section area, and surface area instantly.
Three short formulas, all built from the side length s, the prism length L, and a fixed pentagon constant.
volume = ¼√(5(5 + 2√5)) × s² × LThe cross-section area is the area of one regular pentagon, ¼√(5(5 + 2√5)) × s² — about 1.720477 × s². The volume is simply that area multiplied by the prism length L: a prism holds its cross-section stretched along its length. The total surface area is the two pentagonal ends plus the five rectangular sides, 2 × area + 5 × s × L.
Suppose you have a regular pentagonal prism with a side of 4 and a length of 10.
Cross-section area
¼√(5(5 + 2√5)) × 4² = 1.720477 × 16 = 27.527638 square units — one pentagonal end.
Volume
27.527638 × 10 = 275.276384 cubic units — the space inside.
Surface area
2 × 27.527638 + 5 × 4 × 10 = 255.055277 square units — two ends plus five rectangles.
The three outputs answer three different everyday questions. The volume (about 275.276384 cubic units for s = 4, L = 10) is how much the prism holds — the material in a pentagonal bar, the capacity of a prism-shaped container. The cross-section area (about 27.527638 square units) is the size of one pentagonal end, which is the same all the way along the prism; this is what you would see if you sliced straight through. The surface area (about 255.055277 square units) is the full outside skin — the two pentagonal caps plus the five rectangular sides — useful for wrapping, painting, or estimating material. A handy check: the volume always equals the cross-section area times the length, so doubling the prism length doubles the volume but leaves the cross-section unchanged. The pentagon constant ¼√(5(5 + 2√5)) ≈ 1.720477 is the thread tying it together — it links any pentagon side to its area, and so to the whole prism.
The formulas are exact, but a couple of practical points are worth keeping in mind.
Regular pentagons and consistent units
These formulas describe a right prism with a regular pentagonal cross-section — all five sides equal and the two ends perpendicular to the length. An irregular pentagon, an oblique prism (ends not square to the length), or rounded edges will differ from the computed value. The side and length are also unit-agnostic, so the answers are only meaningful if you keep one unit throughout: a side and length in centimetres give a volume in cubic centimetres and a surface area in square centimetres, never a mix.