Regular Hexagonal Prism Volume Formula

This calculator works out the volume of a regular hexagonal prism from the side length of the hexagon and the prism height. Enter those two values, and you get the volume in cubic units, plus the base area and lateral surface area for reference. The formula is V = (3√3/2) × a² × h, where a is the side length of the regular hexagon and h is the prism height.

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 (six equal sides and six equal angles). The area of a regular hexagon with side length a is:

A_base = (3√3 / 2) × a²

So the volume is:

V = A_base × h = (3√3 / 2) × a² × h

Where:

• V = volume (cubic units, e.g., cm³, m³, in³)
• a = side length of the regular hexagon (linear units, e.g., cm, m, in)
• h = height of the prism (same linear unit as a)
• √3 ≈ 1.73205

The factor (3√3/2) ≈ 2.598 is constant for any regular hexagon. You can also think of the hexagon as six equilateral triangles, each with area (√3/4)a², so the total base area is 6 × (√3/4)a² = (3√3/2)a².

Worked Example

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

1. First, compute the base area: A_base = (3√3/2) × 5² = (3 × 1.73205 / 2) × 25 = 2.59808 × 25 = 64.9519 cm².
2. Then multiply by the height: V = 64.9519 × 10 = 649.519 cm³.
3. Rounded to two decimal places, V = 649.52 cm³.

The calculator shows 649.52 cm³, matching this step-by-step result. The lateral surface area (the six rectangular faces) is 6 × a × h = 6 × 5 × 10 = 300 cm², and the total surface area would be lateral plus two bases: 300 + 2 × 64.95 = 429.90 cm².

When to Use This Formula

Use this formula whenever you need the volume of a prism whose base is a regular hexagon and whose sides are perpendicular to the base (a right prism). This is common in:

• Packing problems – hexagonal boxes, honeycomb-like storage, or hexagonal tiles.
• Engineering – hexagonal nuts, bolts, or structural columns.
• Architecture and design – hexagonal pillars, planters, or furniture.
• Math homework and geometry problems.

If the prism is oblique (leaning), the volume is still base area × perpendicular height, not the slanted edge length. If the hexagon is not regular (sides or angles unequal), this formula does not apply; you would need to compute the actual base area separately.

How to Read the Result

The volume is always in cubic units. If you enter side length and height in centimeters, the volume is in cubic centimeters (cm³). If you use inches, you get cubic inches (in³). For practical purposes, you may want to convert to liters or gallons:

• 1 cm³ = 1 milliliter (mL)
• 1,000 cm³ = 1 liter
• 1 m³ = 1,000 liters
• 1 in³ ≈ 0.004329 US gallons
• 1 ft³ ≈ 7.4805 US gallons

For example, a prism with a = 5 cm and h = 10 cm holds 649.52 mL, or about 0.65 liters. If you need the capacity in gallons, divide the cubic inches by 231 (US liquid gallon).

Common Mistakes and How to Avoid Them

• Using the wrong base area: Some people use the formula for a hexagon’s area with apothem (perimeter × apothem / 2). That works too, but you must know the apothem. The side-length formula is simpler when you only have the side.
• Mixing units: If the side is in inches and the height in feet, convert one first. Always use the same unit for both dimensions.
• Forgetting the √3 factor: The base area is about 2.598 times the square of the side, not 2.598 times the side itself. Squaring the side is essential.
• Using the diameter instead of the side: In a regular hexagon, the distance across opposite vertices (the long diagonal) is 2a. If you have that, divide by 2 to get the side.
• Rounding too early: If you round √3 to 1.73, the error compounds. Use 1.73205 or keep the exact expression until the final step.

Alternatives and Related Formulas

If you have the apothem (the distance from the center to the midpoint of a side), the base area is also A = (perimeter × apothem) / 2 = (6a × apothem) / 2 = 3a × apothem. Then multiply by height.

For a hexagonal prism with an irregular hexagon base, break the base into triangles or use a general polygon area formula. The volume is still base area × height.

If you need the surface area, add the lateral area (6 × a × h) to twice the base area. The diagonal of a side face is √(a² + h²), useful for certain packing or structural calculations.

For other prisms, the same volume principle applies: volume = base area × height. For a cylinder, the base is a circle (πr²); for a triangular prism, it’s (1/2) × base × triangle height.

Frequently Asked Questions

What is the volume of a regular hexagonal prism with side 4 and height 9?

Using V = (3√3/2) × 4² × 9 = 2.59808 × 16 × 9 = 374.12 cubic units (rounded to two decimals).

How do I find the side length if I know the volume and height?

Rearrange the formula: a = √(V / (h × (3√3/2))). For example, if V = 1000 cm³ and h = 10 cm, then a = √(1000 / (10 × 2.59808)) = √(38.49) ≈ 6.20 cm.

Does the formula work for any hexagonal prism?

No, it only works for a regular hexagon (all sides and angles equal). For an irregular hexagon, you must compute the base area using other methods, then multiply by height.

What is the difference between a hexagonal prism and a hexagonal pyramid?

A prism has two parallel hexagonal bases and rectangular side faces; a pyramid has one hexagonal base and triangular sides meeting at a point. The volume of a hexagonal pyramid is (1/3) × base area × height.

How many faces, edges, and vertices does a hexagonal prism have?

It has 8 faces (2 hexagons + 6 rectangles), 18 edges, and 12 vertices. This is a standard property of a hexagonal prism.

Can I use this formula for a honeycomb cell?

Yes, honeycomb cells are roughly hexagonal prisms. If you measure the side length and depth of a cell, you can estimate its volume. However, real cells have slightly rounded corners, so the result is an approximation.

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in a | Side length | cm | 5 | 0
in h | Height | cm | 10 | 0
out vol | Volume | cm³ | (3 * sqrt(3) / 2) * a^2 * h | 2
out base | Base area | cm² | (3 * sqrt(3) / 2) * a^2 | 2
out lat | Lateral surface area | cm² | 6 * a * h | 2