Volume of a Prism: Formula & Examples

This calculator works out the volume of any prism when you enter the area of its cross-section (base) and its length (height). You get the volume in cubic units, along with a step-by-step breakdown of the formula so you can check the working yourself.

Formula

The volume of a prism is calculated with the simple formula:

Volume = Base Area × Length (or Height)

In symbols: V = A × l

• V = volume of the prism, in cubic units (cm³, m³, in³, etc.)
• A = area of the cross-section (the shape you keep slicing through), in square units (cm², m², in², etc.)
• l = length (or height) of the prism, measured perpendicular to the cross-section, in linear units (cm, m, in, etc.)

The base area is the area of the shape that is repeated along the prism. For a rectangular prism, that is length × width. For a triangular prism, it is ½ × base × triangle height. For a cylinder (which is a circular prism), it is π × radius².

Worked Example

Using the calculator defaults: base area = 20 cm², length = 10 cm.

1. Write down the formula: V = A × l
2. Substitute the values: V = 20 × 10
3. Multiply: V = 200

The volume is 200 cm³. This matches the calculator output. If the base area were measured in m² and the length in m, the result would be m³.

What Exactly Is a Prism?

A prism is a 3D shape with two identical, parallel faces (the bases) and other faces that are parallelograms (usually rectangles). The cross-section is the same all the way through. Common prisms include:

• Rectangular prism (like a box or a brick)
• Triangular prism (like a Toblerone box)
• Cylinder (a prism with circular bases)
• Pentagonal prism, hexagonal prism, and so on

If the shape tapers to a point (like a pyramid or cone), it is not a prism – those use a different formula (⅓ × base area × height).

How to Find the Base Area

The hardest part is usually working out the area of the cross-section. Here are the common ones:

• Rectangle: width × height
• Triangle: ½ × base × height
• Circle: π × radius²
• Trapezium: ½ × (a + b) × height
• Regular polygon: (perimeter × apothem) / 2

Once you have that area, just multiply by the length of the prism. For example, a triangular prism with a base area of 6 cm² and a length of 8 cm has a volume of 48 cm³.

Common Mistakes and How to Avoid Them

Several errors trip up students when calculating prism volume:

• Using the wrong base area: Make sure you use the area of the cross-section, not the surface area of one side.
• Mixing units: If the base area is in cm² and the length is in m, convert one to the other first. The volume unit will be the cube of the unit you used.
• Confusing length with height: For a prism lying on its side, the “height” is the length of the prism, not the height of the base triangle.
• Forgetting the ½ for triangles: The area of a triangle is ½ × base × height, not base × height.
• Rounding too early: If you round the base area before multiplying by the length, you can get a noticeably different answer.

Real-Life Applications

Knowing how to find the volume of a prism is useful in many everyday situations:

• Concrete for a driveway: A rectangular slab is a rectangular prism. Volume = length × width × thickness.
• Water in a fish tank: A tank is a rectangular prism. Volume = length × width × height (then convert to litres).
• Packaging: Boxes and cartons are rectangular prisms; volume tells you how much they hold.
• Building materials: Triangular prisms are used for roof trusses and ramps; volume helps estimate material needs.
• Science experiments: Measuring the volume of irregular objects often involves water displacement in a rectangular container.

Limits of the Formula

The formula V = A × l only works for prisms – shapes with a constant cross-section. It does not apply to:

• Pyramids and cones: These taper to a point; use V = ⅓ × base area × height.
• Spheres: Use V = 4/3 × π × radius³.
• Irregular shapes: If the cross-section changes, you cannot use this simple formula.

Also, the formula assumes the prism is “right” – that the length is perpendicular to the base. For an oblique prism (leaning over), the length must be the perpendicular height, not the slanted edge length. In most school problems, the prism is right, but it is worth checking.

Frequently Asked Questions

What is the difference between a prism and a pyramid?

A prism has two identical bases and a constant cross-section, while a pyramid has one base and all faces meet at a single point (apex). The volume formulas differ: prism is base area × length, pyramid is ⅓ × base area × height.

How do I find the volume of a triangular prism?

First, find the area of the triangular cross-section (½ × base of triangle × height of triangle). Then multiply that by the length of the prism. For example, a triangular base with area 12 cm² and a prism length of 5 cm gives a volume of 60 cm³.

Is a cylinder a prism?

Technically, a cylinder is not a polyhedron, but it behaves like a prism because its cross-section (a circle) is constant. The volume formula is the same: base area (πr²) × height. Many textbooks treat cylinders as circular prisms for volume calculations.

Can I use this formula for any shape with a constant cross-section?

Yes, as long as the cross-section is the same all the way along the length and the length is perpendicular to the cross-section. This includes prisms with any polygonal base and also cylinders. If the cross-section changes, you cannot use this simple formula.

Why do I get different answers if I use cm and m?

Because the units are different. If you measure the base area in cm² and the length in cm, the volume is in cm³. If you measure in m, the volume is in m³. 1 m³ = 1,000,000 cm³. Always keep the same unit system throughout the calculation.

How do I convert volume to litres?

1 litre = 1000 cm³. So divide the volume in cm³ by 1000 to get litres. For example, a tank with volume 20,000 cm³ holds 20 litres. For cubic metres, 1 m³ = 1000 litres.

Related

• Stop AI Companies Using Your Chats to Train Models
• Kraken Biome Chameleon Locations: What’s Actually Known
• Emergency Cash for Unstable Markets: How Much to Keep

in a | Base area | cm² | 20 | 0
in l | Length (height) | cm | 10 | 0
out v | Volume | cm³ | a * l | 2