Pentagonal Prism Volume Calculator

A pentagonal prism has two parallel regular pentagon ends and five rectangular side faces connecting them. Decorative architectural columns, five-sided planters and a good deal of specialist packaging use this shape.

V = (1/4) × √(25 + 10√5) × a² × L ≈ 1.7204774 × a² × L

Where a is the edge length of the pentagon (all five sides equal) and L is the prism length (height). The leading constant comes from the regular pentagon area formula:

A_pentagon = (1/4) × √(25 + 10√5) × a² ≈ 1.7204774 × a²

Worked example: a five-sided planter A cedar planter with a = 25 cm pentagonal cross-section, 40 cm deep. A_pentagon = 1.7205 × 625 ≈ 1,075.3 cm². V = 1,075.3 × 40 ≈ 43,012 cm³, which is 43 litres of potting soil. Compost comes in 50-litre bags, so one bag fills it with a little to spare.

The outside width matters as much as the volume when you are deciding where it goes. The apothem is 17.2 cm, so it measures 34.4 cm across the flats and 42.5 cm corner to corner. Plan the gap against a wall from the larger number.

Where pentagonal prisms show up:

• Architectural columns in some Art Deco and Postmodern buildings.
• Planters, bollards and bin surrounds where five sides read as less institutional than four.
• Packaging. Artisan candle, tea and spirits boxes, chosen precisely because the shape is unusual on a shelf.
• Dice, indirectly. The familiar d10 is a pentagonal trapezohedron, the dual of a pentagonal antiprism, not a prism. True pentagonal-prism dice exist only as novelties.
• Cookie cutters and candy moulds in a five-pointed or five-sided form.

Why you will not find this shape in a crystal

Five-fold symmetry is impossible in an ordinary crystal, and that is a theorem rather than an accident. A periodic lattice can only have 2, 3, 4 or 6-fold rotational symmetry, because those are the only rotations that map a repeating grid onto itself. Pentagons simply cannot tile the plane: three of them round a point leave a 36° gap and four overlap. So apatite, beryl and quartz all form hexagonal prisms, never pentagonal ones. (Quasicrystals, discovered in 1982, do show five-fold symmetry, and they get away with it by not being periodic at all.)

The same tiling failure is why bees build hexagonal comb and why pencils are hexagonal rather than pentagonal: hexagons pack against each other with no gaps, and pentagons waste the space between.

Pentagon area shortcut:

For quick mental math:

• a = 1 cm: A ≈ 1.72 cm²
• a = 2 cm: A ≈ 6.88 cm², four times the 1 cm figure, because area is quadratic in the side
• a = 5 cm: A ≈ 43 cm²

Multiply by prism length L to get volume.

Comparing to a hexagonal prism with the same edge length:

A hexagonal prism has cross-section (3√3/2) × a² ≈ 2.598 × a², so for the same edge length it holds about 51% more than a pentagonal one.

Measure them by the same across-flats width instead and the result flips. In terms of the inradius r, a pentagon covers 5r²·tan(36°) = 3.633r² while a hexagon covers 6r²·tan(30°) = 3.464r², so the pentagon is about 4.9% LARGER. Fewer sides bulge further past the inscribed circle at each corner, which is the same reason a triangle of a given inradius beats them both.

Which comparison you want depends on what is fixed. If you are cutting from stock of a known width, compare by across-flats. If you are buying edge trim, compare by edge length.

Sanity check:

• L = 0: V = 0. ✓
• a = 0: V = 0. ✓
• For a = 1, L = 1: V ≈ 1.7204774. (Unit pentagonal prism.)