Volume of a Regular Prism: Formula & Examples

To find the volume of a regular prism, multiply the area of its base by its height: V = Bh. You only need two measurements — the base’s side length (and number of sides) and the prism’s height — and the calculator below does the rest.

Formula

For any prism, volume equals base area times height: V = B × h. For a regular prism, the base is a regular polygon (all sides and angles equal), so its area is:

B = (n × s²) / (4 × tan(π / n))

Where:

• V = volume (cubic units, e.g., cm³, m³, in³)
• B = area of one base (square units)
• h = height of the prism, perpendicular distance between the two bases (linear units)
• n = number of sides of the regular polygon
• s = length of one side of the polygon (linear units)

This formula works for any regular polygon base: triangular (n=3), square (n=4), pentagonal (n=5), hexagonal (n=6), and so on. The tangent function uses radians in the calculator, but the formula is the same in degrees: tan(180°/n).

Worked Example

Using the calculator defaults: a regular hexagonal prism (n=6) with side length s=5 cm and height h=10 cm.

1. Compute the base area: B = (6 × 5²) / (4 × tan(π/6)) = (6 × 25) / (4 × 0.57735) = 150 / 2.3094 = 64.95 cm².
2. Multiply by height: V = 64.95 × 10 = 649.5 cm³.

The calculator rounds to 2 decimals, showing base area 64.95 cm² and volume 649.50 cm³. If you use a cube (n=4, s=5, h=5), the formula gives (4 × 25)/(4 × tan(π/4)) = 100/4 = 25 cm² base, times height 5 = 125 cm³, which matches the familiar side³.

When to Use This Formula

Use this whenever you have a prism with a regular polygon base and a uniform height. Common real-world items include hexagonal bolts, triangular wooden beams, square posts, and pentagonal glass vases. The height must be measured perpendicular to the base — not along a slanted edge. If the prism is oblique (leaning), the volume still uses the perpendicular height, not the slant length.

For irregular bases (like a trapezoid or a random quadrilateral), you cannot use the regular polygon formula. Instead, compute the actual base area using the appropriate shape formula (e.g., trapezoid area) and multiply by height.

How to Measure Correctly

• Measure the side length of the base accurately — if the polygon is not regular, measure each side and use a different area formula.
• Measure the height straight up from the base to the top, at a right angle to the base.
• Keep all units consistent. If side length is in inches, height must be in inches, and volume will be in cubic inches.
• For a cube, all edges are equal, so height equals side length. For other prisms, height is a separate measurement.

Common Mistakes and Misconceptions

One frequent error is using the slant height of an oblique prism instead of the perpendicular height. Another is forgetting to divide by the tangent term when the base is not a square. For a square base (n=4), tan(π/4)=1, so the formula simplifies to s², but for other polygons the tangent term matters. Also, mixing units (e.g., side in cm and height in m) will produce a volume in a nonsense unit. Always convert to the same unit first.

Some people confuse volume with surface area. Volume measures the space inside, in cubic units; surface area measures the total exterior, in square units. The calculator above only computes volume.

Limitations of the Formula

The formula assumes a perfect regular polygon and a prism with parallel, congruent bases. Real objects may have manufacturing tolerances or irregular shapes. For a cylinder (a prism with a circular base), the base area is πr², which is the limit of the regular polygon formula as n approaches infinity. For a cone or pyramid, the volume is one-third of the base area times height, so this formula does not apply.

If the prism is truncated or has a non-uniform cross-section, you need calculus or a different geometric approach.

Frequently Asked Questions

What is a regular prism? A prism with two identical, parallel bases that are regular polygons (all sides and angles equal) and rectangular side faces. Examples include cubes, regular triangular prisms, and hexagonal prisms.

Does the formula work for any prism? The V = Bh part works for any prism, but the base area formula (n s²)/(4 tan(π/n)) only works for regular polygons. For irregular bases, you must find the actual base area separately.

What if I only know the apothem (distance from center to side midpoint)? Then base area = (perimeter × apothem)/2. For a regular polygon, perimeter = n × s. Multiply that area by height to get volume.

How do I convert cubic centimeters to liters? 1 liter = 1,000 cm³. So the example volume of 649.5 cm³ equals 0.6495 liters. For gallons, 1 US gallon ≈ 3,785.41 cm³.

Why does the formula use tangent? The regular polygon is divided into n congruent isosceles triangles. The height of each triangle is s/(2 tan(π/n)), and the area of one triangle is (1/2) × s × that height. Multiplying by n gives the formula.

Can I use this for a cylinder? No, a cylinder has a circular base, not a polygon. Use V = πr²h instead. However, as n gets very large, the regular polygon formula approaches the cylinder formula.

Related

• Minutes to Hours: Divide by 60? Yes, Here’s How
• Minutes to Hours: Formula and Converter
• Google Minutes to Hours? Quick Conversion & Guide

in s | Side length | cm | 5 | 0
in n | Number of sides | - | 6 | 3
in h | Height | cm | 10 | 0
out area | Base area | cm² | (n * s^2) / (4 * tan(pi / n)) | 2
out volume | Volume | cm³ | area * h | 2