Hexagonal Prism Volume: Formula & Calculator

This calculator works out the volume of a regular hexagonal prism from the side length of the hexagon and the height (or length) of the prism. Enter those two measurements in any consistent unit, and you get the volume in cubic units (e.g., cubic inches, cubic centimeters).

Formula

The volume of any prism is the area of its base multiplied by its height. For a regular hexagonal prism, the base is a regular hexagon (all six sides equal). The formula is:

V = (3√3 / 2) × s² × h

• V = volume (cubic units, e.g., in³, cm³, m³)
• s = side length of the hexagonal base (same linear unit as h)
• h = height (or length) of the prism, perpendicular to the base (same linear unit as s)
• (3√3 / 2) × s² = area of a regular hexagon (square units)

The factor 3√3/2 ≈ 2.598 is a constant. It comes from dividing the hexagon into six equilateral triangles, each with area (√3/4)s².

Worked Example

Using the default values from the calculator: side length s = 5 in, height h = 12 in.

1. Compute the base area: (3√3 / 2) × 5² = 2.598 × 25 = 64.95 in² (rounded to two decimals).
2. Multiply by height: 64.95 × 12 = 779.4 in³.

The calculator returns 779.40 in³ for the volume and 64.95 in² for the base area, matching the steps above.

When to Use This Formula

Use this formula whenever you have a prism with a regular hexagonal base and straight, parallel sides. Common real-world items include hexagonal bolts, nuts, pencils, glass vases, and some building columns. If the hexagon is irregular (sides not equal), this formula does not apply—you must find the actual base area another way.

The formula works for any unit system. If you mix units (e.g., side in inches, height in feet), you must convert first. The result is always in cubic units of the same system.

How to Read the Result

The volume tells you the total space inside the prism. For practical uses:

• If you need capacity in gallons or liters, convert cubic inches (1 gal = 231 in³) or cubic centimeters (1 L = 1000 cm³).
• If you need weight, multiply the volume by the material density (e.g., steel ≈ 490 lb/ft³, water ≈ 62.4 lb/ft³).
• If you need surface area, that is a different calculation—volume alone does not give it.

Common Mistakes and Misconceptions

• Using the apothem instead of side length: The formula uses side length s. If you know the apothem (a, the distance from center to midpoint of a side), the base area is (1/2) × perimeter × apothem = (1/2) × 6s × a = 3sa. Then volume = 3sa × h.
• Mixing units: Always convert to one unit before calculating. For example, if s is in cm and h is in m, convert h to cm (multiply by 100) or s to m (divide by 100).
• Forgetting the constant: The factor 3√3/2 is not optional. Some people mistakenly use 2.6 or 2.598—use the exact value for precision.
• Using the diameter of a circumscribed circle: The side length of a regular hexagon equals the radius of its circumscribed circle (the distance from center to a vertex). If you have the diameter, divide by 2 to get s.

Alternatives and Related Formulas

If you have different measurements, you can still find the volume:

• Given apothem a and side s: Volume = 3 × a × s × h (since base area = 3as).
• Given circumradius R (same as s): Use the same formula with s = R.
• Given base area A directly: Volume = A × h.
• For irregular hexagonal bases: Divide the base into triangles or use surveyor’s formula, then multiply by h.

Limits of the Formula

This formula assumes a perfect regular hexagon and a right prism (sides perpendicular to the base). If the prism is oblique (slanted), the volume is still base area × perpendicular height, but you must use the vertical height, not the slanted edge length. For very large or very small values, rounding errors can accumulate; keep intermediate steps with more decimals.

Frequently Asked Questions

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

Use the formula V = 3 × a × s × h, where a is the apothem and s is the side length. If you don’t know s, you can relate them: for a regular hexagon, s = 2a / √3. So V = 3 × a × (2a/√3) × h = 2√3 × a² × h.

What if my hexagon is not regular?

You must calculate the area of the irregular hexagon separately (e.g., by splitting it into triangles) and then multiply by the height. The simple formula with s² only works for regular hexagons.

Can I use this formula for a hexagonal pyramid?

No. A pyramid has a different volume formula: V = (1/3) × base area × height. For a regular hexagonal pyramid, that would be V = (1/3) × (3√3/2) × s² × h.

How do I convert cubic inches to gallons?

Divide the volume in cubic inches by 231. For example, 779.4 in³ ÷ 231 ≈ 3.37 gallons. To convert cubic centimeters to liters, divide by 1000.

Why is the constant 3√3/2?

A regular hexagon consists of six equilateral triangles. Each triangle has area (√3/4)s². Multiplying by 6 gives (6√3/4)s² = (3√3/2)s².

Does the height have to be perpendicular?

Yes. For a right prism, the height is the perpendicular distance between the two bases. If the prism is slanted, use the vertical height, not the slanted edge length.

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in s | Side length | in | 5 | 0
in h | Height | in | 12 | 0
out V | Volume | in³ | (3 * sqrt(3) / 2) * s^2 * h | 2
out base_area | Base area | in² | (3 * sqrt(3) / 2) * s^2 | 2