Pentagonal Prism Volume Formula & Calculator

This calculator works out the volume of a pentagonal prism from the side length of its regular pentagonal base and the prism’s height. Enter those two values and you get the volume in cubic units, plus the base area and surface area for reference. The formula used is the standard one for a regular pentagonal prism: V = (1/4) × √(5(5+2√5)) × s² × h.

Formula

The volume of a pentagonal prism is the area of its pentagonal base multiplied by the prism’s height (the distance between the two bases). For a regular pentagon (all sides and angles equal), the base area is:

A_base = (1/4) × √(5(5+2√5)) × s²

So the volume is:

V = A_base × h = (1/4) × √(5(5+2√5)) × s² × h

• V = volume (cubic units, e.g., cm³, m³, in³)
• s = side length of the regular pentagon (linear units, e.g., cm, m, in)
• h = height of the prism (same linear units as s)
• √(5(5+2√5)) ≈ 6.8819 (a constant that comes from pentagon geometry)

If your pentagon is not regular (irregular), you cannot use this simple formula. In that case, you must find the actual area of the pentagon (e.g., by dividing it into triangles) and multiply by the height.

Worked Example

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

1. Calculate the base area:
   A_base = (1/4) × √(5(5+2√5)) × (5)²
   = 0.25 × 6.8819 × 25
   = 43.0119 cm² (rounded to 43.01 cm²)
2. Multiply by the height:
   V = 43.0119 × 10 = 430.119 cm³
3. Rounded to two decimals: V = 430.12 cm³

The calculator shows the same result: 430.12 cm³. The base area shows 43.01 cm², and the surface area (5 × s × h + 2 × base area) = 5×5×10 + 2×43.01 = 250 + 86.02 = 336.02 cm².

When to Use This Formula

Use this formula whenever you need the volume of a prism whose base is a regular pentagon. Common real-world examples include:

• Pentagonal building columns or pillars
• Pencils with a pentagonal cross-section (before sharpening)
• Certain nuts, bolts, or machine parts
• Architectural elements, decorative blocks, or packaging
• Math homework and geometry problems

The key is that the two bases are parallel and congruent, and the sides are rectangles (a right prism). If the prism is oblique (leaning), the volume is still base area × height, but the height is the perpendicular distance between bases, not the slant length.

How to Read the Result

The volume is always in cubic units. If you input side length in centimeters and height in centimeters, the volume is in cubic centimeters (cm³). If you use inches, you get cubic inches (in³). The base area is in square units (cm², in², etc.) and the surface area is also in square units.

For practical purposes, 1 cm³ = 1 milliliter. So 430.12 cm³ is exactly 430.12 mL. To convert to liters, divide by 1000: 0.430 L. For cubic meters, 1 m³ = 1000 liters. If you need gallons, 1 US gallon ≈ 3,785.41 cm³, so 430.12 cm³ ≈ 0.1136 US gallons.

Common Mistakes and Misconceptions

• Using the apothem incorrectly: Some formulas use the apothem (distance from center to midpoint of a side). The formula with side length is different. If you have the apothem a, the base area is (5/2) × s × a. Don’t mix the two.
• Confusing side length with radius: The side length is not the distance from the center to a vertex (circumradius). For a regular pentagon, circumradius R = s / (2 sin 36°). Using R instead of s in the volume formula gives a wrong answer.
• Forgetting to square the side length: The formula has s². Forgetting the square is a common error that makes the volume too small.
• Unit mix-ups: If side length is in meters and height in centimeters, convert one first. The volume will be in the cube of the unit you use for both.
• Irregular pentagon: The formula only works for regular pentagons. For irregular ones, you need the actual base area.

Limitations of the Formula

This formula assumes a perfectly regular pentagon and a right prism. Real objects might have slight manufacturing tolerances, but the formula is accurate for theoretical calculations. For very large or very small prisms, the same formula works; there is no limit. The constant √(5(5+2√5)) is exact, but in practice you’ll round it to 6.8819 or use the calculator’s precision.

If the pentagon is irregular, you cannot use a single formula. You would need to decompose the pentagon into triangles, find each triangle’s area, sum them, and multiply by the height. Alternatively, use a polygon area calculator that accepts vertex coordinates.

Frequently Asked Questions

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

A prism has two identical parallel bases connected by rectangular faces. A pyramid has one pentagonal base and triangular faces that meet at a single apex. The volume of a pentagonal pyramid is (1/3) × base area × height, not base area × height.

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

If you know the apothem (a) and the side length (s), the base area is (5/2) × s × a. If you only know the apothem and not the side length, you can find s from the apothem: s = 2a × tan(36°). Then use the volume formula with that s.

Can I use this formula for a pentagonal prism with a non-regular base?

No. The formula only works for a regular pentagon. For an irregular pentagonal base, you must calculate the actual area of the pentagon (e.g., by triangulation) and multiply by the height.

What units should I use?

Any consistent units. If you mix units (e.g., side in cm and height in m), convert one first. The volume will be in cubic units of the unit you use for both.

How do I find the surface area of a pentagonal prism?

Surface area = 5 × s × h + 2 × base area. The first term is the area of the five rectangular side faces, and the second term is the two pentagonal bases. The calculator provides this as an output.

Why is the constant √(5(5+2√5)) in the formula?

It comes from the trigonometric derivation of the area of a regular pentagon. For a regular pentagon, the area equals (5/4) × s² × cot(36°), and cot(36°) simplifies to √(5(5+2√5))/5, which leads to the constant you see.

Related

• IST to PST Time Converter: How It Works
• BB per 9 Calculator: Formula, Examples & How to Use
• IST to EST to UTC Converter Online

in s | Side length | cm | 5 | 0
in h | Height | cm | 10 | 0
out V | Volume | cm³ | (1/4) * sqrt(5*(5+2*sqrt(5))) * s^2 * h | 2
out A_base | Base area | cm² | (1/4) * sqrt(5*(5+2*sqrt(5))) * s^2 | 2
out A_surf | Surface area | cm² | 5 * s * h + 2 * ((1/4) * sqrt(5*(5+2*sqrt(5))) * s^2) | 2