Volume of Hexagonal Prism: Formula & Calculator

This calculator works out the volume of a regular hexagonal prism from the side length of the hexagon and the height of the prism. You enter those two measurements, and you get the volume in cubic units, plus the base area and the total surface area for reference. The formula used is the standard one for a regular hexagon base: V = (3√3/2) × a² × h.

Formula

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

Base area = (3√3 / 2) × a²

where a is the side length of the hexagon. So the full volume formula is:

V = (3√3 / 2) × a² × h

• V = volume (cubic units, e.g., cm³, m³, in³)
• a = side length of the hexagonal base (linear units, e.g., cm, m, in)
• h = height (or length) of the prism, perpendicular to the base (same linear unit as a)

The constant (3√3 / 2) is approximately 2.598. It comes from dividing the regular hexagon into six equilateral triangles. Each equilateral triangle with side a has area (√3/4)a², and six of them give (6 × √3/4)a² = (3√3/2)a².

Worked Example

Using the default values from the calculator: side length a = 5 cm, height h = 10 cm.

1. Compute the base area: (3√3 / 2) × 5² = 2.598 × 25 = 64.95 cm² (rounded to 2 decimals).
2. Multiply by the height: 64.95 × 10 = 649.5 cm³.

The calculator gives 649.52 cm³ (the small difference is from rounding the constant earlier; the calculator uses the exact expression). So the volume is 649.52 cm³.

When to Use This Calculation

You need the volume of a hexagonal prism whenever you have a solid object with two identical hexagonal faces and six rectangular side faces. Common real-world cases:

• Hexagonal nuts and bolts (volume of metal, for weight estimates)
• Hexagonal columns or pillars (concrete or steel volume)
• Hexagonal glass tanks or containers (water capacity)
• Hexagonal pencils (volume of wood and graphite)
• Architectural elements like hexagonal skylights or planters

The same formula applies whether the prism is standing on its hexagon (vertical) or lying on its side (like a long hexagonal bar). The “height” is always the distance between the two hexagonal faces.

How to Read the Result

The volume is always in cubic units. If you entered side and height in centimeters, the volume is in cubic centimeters (cm³). For meters, it’s cubic meters (m³); for inches, cubic inches (in³).

To convert to a more practical unit:

• 1 cm³ = 1 milliliter (mL). So 649.52 cm³ = 649.52 mL = 0.64952 liters.
• 1 m³ = 1,000 liters.
• 1 in³ = 16.387 cm³.
• 1 ft³ = 28.317 liters.

If you need the mass of the prism, multiply the volume by the density of the material. For example, steel has a density around 7,850 kg/m³. A hexagonal steel bar with a volume of 0.0006495 m³ would weigh about 5.1 kg.

Common Mistakes and How to Avoid Them

• Using the wrong “side”: The formula uses the side length of the hexagon, not the distance across flats (the width between two parallel sides). If you only know the distance across flats (call it d), the side length is a = d / √3. Many real hexagonal objects are specified by across-flats, so check your measurement.
• Mixing units: Always use the same unit for side and height. If side is in cm and height in m, convert one first. The calculator assumes consistent units.
• Confusing base area with volume: The base area is a two-dimensional quantity (cm²). The volume is three-dimensional (cm³). Don’t stop at the base area.
• Using the apothem incorrectly: The apothem (distance from center to midpoint of a side) is related to the side by apothem = (√3/2) × a. Some formulas use apothem × perimeter / 2 for area, which is also correct, but the side-based formula is simpler when you have the side length.

Limits of the Formula

This formula applies only to a regular hexagonal prism – one where all six sides of the hexagon are equal and all angles are 120°. If the hexagon is irregular (sides or angles differ), you cannot use this single formula. Instead, you would need to calculate the area of that specific hexagon (e.g., by dividing it into triangles) and then multiply by the height.

Also, the prism must be “right” – meaning the height is perpendicular to the base. If the prism is oblique (leaning), the volume is still base area × perpendicular height, but you must use the vertical height, not the slant length along the side.

For very large or very small values, rounding can become significant. The calculator rounds to 2 decimal places, which is fine for most practical purposes.

Frequently Asked Questions

Q: How do I find the volume if I only know the apothem (distance from center to side)?
A: The apothem r of a regular hexagon is related to the side length by a = 2r / √3. Substitute that into the volume formula: V = (3√3/2) × (2r/√3)² × h = (2√3) × r² × h. So you can use V = 2√3 × r² × h.

Q: What if my hexagon is not regular?
A: The formula above does not apply. You must calculate the area of the irregular hexagon (e.g., by breaking it into triangles or using coordinates) and then multiply by the height. There is no single simple formula for an arbitrary hexagon.

Q: Does the volume change if I rotate the prism?
A: No. Volume is a fixed property of the solid. Rotating the prism in space does not change its volume. The height in the formula is always the perpendicular distance between the two hexagonal faces, regardless of orientation.

Q: How do I convert the volume to gallons or liters?
A: Use these conversions: 1 cm³ = 0.001 liters; 1 liter = 0.264172 US gallons; 1 m³ = 264.172 US gallons. For example, 649.52 cm³ = 0.64952 L = 0.1716 US gallons.

Q: Can I use this for a hexagonal column of concrete?
A: Yes. Measure the side length of the hexagonal cross-section and the height of the column. Use the same units (e.g., meters). The volume in m³ times the density of concrete (about 2,400 kg/m³) gives the mass. Always order a little extra for waste.

Q: What is the surface area of a hexagonal prism?
A: The total surface area is the sum of the two hexagonal bases plus the six rectangular side faces: A = 2 × (3√3/2)a² + 6ah = 3√3 a² + 6ah. The calculator provides this as a secondary output.

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in a | Side length | cm | 5 | 0
in h | Height | cm | 10 | 0
out V | Volume | cm³ | (3 * sqrt(3) / 2) * a^2 * h | 2
out A_base | Base area | cm² | (3 * sqrt(3) / 2) * a^2 | 2
out A_surf | Surface area | cm² | 2 * (3 * sqrt(3) / 2) * a^2 + 6 * a * h | 2