Prism Geometry Volume: Formula & Calculator

This calculator works out the volume of any right prism (triangular, rectangular, pentagonal, or any other uniform cross-section) when you enter the area of its base and its height. You get the volume in cubic units, plus the base area and height you entered, so you can check your own working. The formula is simply Volume = Base Area × Height, and this page explains why that works, how to find the base area for common shapes, and where people usually trip up.

Formula

The volume of a prism is the area of its cross-section (the base) multiplied by the prism’s height (or length, if you think of it lying sideways).

V = A × h

• V = volume (cubic units, e.g., cm³, m³, in³)
• A = area of one base face (square units, e.g., cm², m², in²)
• h = height of the prism, measured perpendicular to the base (linear units, e.g., cm, m, in)

This works for any prism – triangular, rectangular, pentagonal, hexagonal, even irregular shapes – as long as the cross-section is the same all the way along. The key is that the base face and the top face are identical, parallel polygons.

Worked Example

Using the default values from the calculator above: a base area of 20 cm² and a height of 10 cm.

1. Write down the formula: V = A × h
2. Plug in the numbers: V = 20 cm² × 10 cm
3. Multiply: V = 200 cm³

The calculator gives 200.00 cm³, which matches. Notice the units: cm² × cm = cm³. That is how you know you are dealing with volume.

What Counts as a Prism?

A prism is a solid with two identical, parallel polygon faces (the bases) and rectangular faces connecting them. The shape of the base can be anything – triangle, square, rectangle, pentagon, hexagon, even a star – as long as it is a polygon and the two bases are congruent and parallel.

If the side faces are rectangles, it is a right prism (the height is perpendicular to the base). If the side faces are parallelograms, it is an oblique prism – and the volume formula still works, but the height is the perpendicular distance between the two bases, not the length of a slanted edge.

Common prisms you will see in school and real life:

• Rectangular prism (a box) – base is a rectangle, volume = length × width × height
• Triangular prism – base is a triangle, often used in optics and architecture
• Pentagonal prism – base is a pentagon, seen in some building designs
• Hexagonal prism – base is a hexagon, common in nuts and bolts

If the base is a circle, it is a cylinder, not a prism (a prism must have a polygon base). But the volume formula for a cylinder is the same idea: V = πr²h.

How to Find the Base Area

The hardest part of most prism volume problems is finding the area of the base, not the volume itself. Here are the formulas for common base shapes:

• Rectangle: A = length × width
• Square: A = side²
• Triangle: A = (base × height) / 2  (the triangle’s own height, not the prism’s height)
• Regular pentagon: A = (1/4) × √(5(5+2√5)) × side², or use A = (perimeter × apothem) / 2
• Regular hexagon: A = (3√3 / 2) × side²
• Any regular polygon: A = (perimeter × apothem) / 2, where the apothem is the distance from the center to the midpoint of a side.

For an irregular polygon, you can often split it into triangles, find each triangle’s area, and add them up.

When to Use This Calculator

This calculator is useful any time you already know the base area and the height. For example:

• You have a triangular prism with a base area of 15 cm² and a length of 8 cm – you want the volume of the prism (which could be a tent, a wedge, or a glass prism).
• You are filling a rectangular pool (a rectangular prism) and know the base area and depth.
• You are working on a geometry homework problem that gives you the base area directly.

If you do not know the base area, you need to calculate it first using the shape’s own formula, then come back here. For a rectangular prism, you can just multiply length × width × height directly – this calculator is more general.

Common Mistakes and How to Avoid Them

Here are the most frequent errors people make when calculating prism volume:

• Mixing up the base height and the prism height. In a triangular prism, the triangle has its own height (used to find the triangle’s area), and the prism has a separate height (the distance between the two triangular faces). They are different numbers.
• Using the wrong units. If the base area is in cm² and the height is in meters, convert one to the other first. The volume will be in cubic units of whichever unit you used.
• Forgetting to halve for triangles. The area of a triangle is (base × height)/2, not base × height. This is the single most common mistake.
• Using the slant height instead of the perpendicular height. For an oblique prism, the height is the perpendicular distance between the bases, not the length of an edge. In a right prism, they are the same.
• Rounding too early. If you round the base area before multiplying by the height, you can lose precision. Keep full precision until the final step.

Limits of the Formula

The formula V = A × h is exact for any prism, but it has limits:

• It only works if the cross-section is constant. If the shape tapers (like a pyramid or cone), you need a different formula (V = (1/3) × base area × height).
• It assumes the base is a flat polygon. If the base is curved (like a cylinder), the formula still works but the shape is not a prism.
• It gives the volume of the solid, not the surface area. Surface area requires adding up the areas of all faces.

For a pyramid, the volume is one-third of the prism volume with the same base and height. That is a common follow-up question – and a common mistake is using 1/3 when the shape is a prism, or forgetting the 1/3 when it is a pyramid.

Frequently Asked Questions

What is the difference between a prism and a pyramid?

A prism has two identical parallel bases and rectangular side faces. A pyramid has one base and triangular side faces that meet at a single point (the apex). The volume of a pyramid is one-third of the volume of a prism with the same base and height.

Does the volume formula work for any prism, even an irregular one?

Yes. As long as the two bases are identical, parallel polygons, the volume is always base area × height. The base area might be harder to calculate for an irregular shape, but the formula itself does not change.

How do I find the volume of a triangular prism if I only know the triangle’s sides?

First, find the area of the triangular base using Heron’s formula: s = (a+b+c)/2, then area = √(s(s-a)(s-b)(s-c)). Then multiply by the prism’s height. If you have a right triangle, you can use (leg1 × leg2)/2 instead.

What units should I use for volume?

Volume is always in cubic units. If you measure in centimeters, the volume is in cubic centimeters (cm³). In meters, it is cubic meters (m³). In inches, cubic inches (in³). 1 m³ = 1,000,000 cm³, and 1 cm³ = 1 milliliter.

Is a cube a prism?

Yes. A cube is a rectangular prism where all six faces are squares. Its volume is side³, which is the same as base area (side²) × height (side).

Can I use this calculator for a cylinder?

Technically, a cylinder is not a prism because its base is a circle, not a polygon. But the volume formula is the same: V = πr²h. You can use this calculator if you enter the circle’s area (πr²) as the base area.

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in a | Base area | cm² | 20 | 0
in h | Height (length) | cm | 10 | 0
out v | Volume | cm³ | a * h | 2
out base_area | Base area | cm² | a | 2
out height | Height | cm | h | 2