Pentagonal Prism Volume Calculator

A pentagonal prism has two parallel regular pentagon ends and five rectangular side faces connecting them. Five-sided pencils, some decorative architectural columns, and quartz-type crystals all have 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 — decorative wooden pencil: A standard wooden pencil is hexagonal, but the rarer pentagonal pencils (Ticonderoga “Soft,” some Eberhard Faber lines) have a = 4 mm pentagonal cross-section, L = 175 mm long. A_pentagon = 1.7205 × 16 ≈ 27.53 mm². V = 27.53 × 175 ≈ 4,817 mm³ ≈ 4.8 cm³ of wood per pencil.

At a wood density of about 0.5 g/cm³ (cedar), that’s ~2.4 g of wood per pencil. Plus the graphite core (a small cylinder) adds another 0.5-1 g.

Where pentagonal prisms show up:

• Decorative pencils (rare — most are hexagonal for cost reasons, since six sides nest more efficiently in production).
• Architectural columns in some Art Deco and Postmodern buildings.
• Some crystal forms in mineralogy — apatite and beryl crystals occasionally develop pentagonal prism habit.
• Game dice (d5 / d10). Pentagonal trapezohedra (d10) are technically derived from pentagonal antiprisms, but pure pentagonal prisms appear in custom dice.
• 5-sided cookie cutters and candy molds.
• Honeycomb-alternative bee hive designs for specialty beekeeping (though hexagonal is the optimal natural form).

Why hexagonal beats pentagonal for packing:

Hexagons tile a plane with zero gaps. Pentagons cannot — no regular pentagon tiling exists. That’s why bees build hexagonal honeycomb cells (maximum volume for minimum wall material), and why hexagonal pencils dominate the market. Pentagons leave wasted space when packed.

Pentagon area shortcut:

For quick mental math:

• a = 1 cm: A ≈ 1.72 cm²
• a = 2 cm: A ≈ 6.88 cm² (4× scaling — area is quadratic in 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 area (3√3/2) × a² ≈ 2.598 × a². So a hexagonal prism holds about 51% more volume than a pentagonal prism of the same edge length and prism length.

For the same outer “size” (same inscribed circle radius), the comparison is different — pentagons and hexagons hold similar volumes when measured by their inscribed circle.

Sanity check:

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