Volume Prism Pentagonal: Formula & Calculator

This calculator works out the volume of a pentagonal prism. You enter the base edge length and the prism height, and you get the volume in cubic units. The volume equals the area of the pentagonal base times the height.

Formula

The volume V of a pentagonal prism is given by:

V = (1/4 × √(5(5 + 2√5)) × a²) × h

• a = length of one side of the regular pentagon base (same unit as volume’s cubic unit)
• h = height of the prism (perpendicular distance between the two pentagonal bases)
• V = volume in cubic units (e.g., m³, cm³, ft³)

The factor (1/4 × √(5(5 + 2√5))) ≈ 1.72048 is the area of a regular pentagon with side length 1. Multiply that by a² to get the base area, then multiply by the height.

Worked Example

Using the default values: base edge a = 4 m, height h = 10 m.

1. Compute the base area: 1.72048 × 4² = 1.72048 × 16 = 27.52768 m².
2. Multiply by height: 27.52768 × 10 = 275.2768 m³.
3. Rounded to two decimals: 275.28 m³.

The calculator gives the same result: 275.28 m³.

When to Use This Calculator

Use this when you need the volume of a prism with a regular pentagonal base – for example, a pentagonal column, a custom tank, a concrete form, or a packaging box. If the base is not a regular pentagon (all sides equal, all angles equal), this formula does not apply directly. For irregular pentagons, you must compute the base area separately (e.g., by triangulation) and then multiply by height.

This is also useful for converting between volume and other quantities: if you know the volume and height, you can solve for the base edge by rearranging the formula.

How to Read the Result

The volume is always in cubic units. If you enter meters, you get cubic meters (m³). If you enter centimeters, you get cubic centimeters (cm³). 1 m³ = 1,000,000 cm³, and 1 ft³ ≈ 0.02832 m³. For practical purposes:

• 1 cubic meter = 1000 liters
• 1 cubic foot ≈ 7.48 US gallons
• 1 cubic inch ≈ 0.004329 US gallons

The calculator also shows the base area separately, which is helpful if you only need the area of the pentagon.

Common Mistakes and Pitfalls

• Using the wrong dimension: The formula uses the side length of the pentagon, not the distance from center to vertex (circumradius) or the apothem. If you have the apothem r, the area is (5/2) × a × r, so volume = (5/2) × a × r × h.
• Mixing units: All dimensions must be in the same unit. Convert everything to meters or everything to feet before calculating.
• Forgetting to square the side: The a² term is essential. Doubling the side length quadruples the base area.
• Confusing height with slant height: The height is the perpendicular distance between the two bases, not the length of a slanted edge (if the prism is oblique). For a right prism, they are the same.

Limitations of the Formula

The formula assumes a regular pentagon and a right prism (bases directly above each other). For an oblique pentagonal prism, the volume is still base area × perpendicular height, but the height is not the same as the edge length. For an irregular pentagon, you must compute the actual base area – the constant 1.72048 does not apply.

Also, the formula gives the mathematical volume. If you are measuring a real object, consider wall thickness, manufacturing tolerances, and whether the shape is truly regular.

Alternatives and Related Shapes

If your base is not a pentagon, the same principle – base area × height – works for any prism. For a triangular prism, use (1/2 × base × triangle height) × prism height. For a hexagonal prism, the base area is (3√3/2) × a². For a cylinder, it’s πr² × height. The key is always: area of the base shape times the perpendicular height.

If you only have the apothem (distance from center to midpoint of a side), you can compute the side length as a = 2 × apothem × tan(π/5) ≈ 1.453 × apothem, then use the formula above.

Frequently Asked Questions

What is the formula for the volume of a pentagonal prism?

The volume is the base area times the height. For a regular pentagon with side length a, the base area is (1/4)√(5(5+2√5)) × a², so V = (1/4)√(5(5+2√5)) × a² × h.

How do I find the volume if I only know the apothem and height?

If you know the apothem r (distance from center to the midpoint of a side), the base area is (5/2) × a × r. But you need the side length a. You can find a from r using a = 2r × tan(36°) ≈ 1.453 × r. Then use the standard formula.

Can I use this for a pentagonal pyramid?

No. A pyramid has a different formula: V = (1/3) × base area × height. This calculator is only for prisms, which have two identical bases.

What if my pentagon is not regular?

For an irregular pentagon, you must compute the actual base area (e.g., by dividing it into triangles) and then multiply by the prism height. The constant 1.72048 only works for regular pentagons.

How do I convert the volume to liters or gallons?

1 cubic meter = 1000 liters. 1 cubic foot = 28.3168 liters ≈ 7.48052 US gallons. So multiply the volume in m³ by 1000 to get liters, or multiply ft³ by 7.48052 to get US gallons.

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in a | Base edge | m | 4 | 0
in h | Height | m | 10 | 0
out V | Volume | m³ | (1/4 * sqrt(5 * (5 + 2 * sqrt(5))) * a^2) * h | 2
out base_area | Base area | m² | 1/4 * sqrt(5 * (5 + 2 * sqrt(5))) * a^2 | 2