Volume Of A Rectangular Prism Calculator

What this calculator does

A rectangular prism is the formal name for a box-shaped solid: six rectangular faces, with a rectangle for its base. It is also called a cuboid. This tool finds the volume, the space inside, from its length, width, and height.

Type the three in and you have the volume. There is no unit setting here, so the result comes in generic cubic units.

Using the calculator

1. Type the width, the length, and the height.
2. Press Calculate.

All three values have to be positive. The volume is reported in cubic units, written unit³, so you can work in whatever unit you like as long as all three are the same.

The formula | volume = length × width × height

The volume of a rectangular prism is:

volume = length × width × height

Multiply the three dimensions. That is the quick version. The next section gives the deeper way to read it, which is the part actually worth carrying away.

The bigger idea: base area times height

A rectangular prism is a rectangle, the base, raised straight up to a height. So its volume is the area of that base times the height:

volume = (base area) × height = (length × width) × height

Here is why that is worth more than the plain formula. The very same rule gives the volume of a whole family of solids. A triangular prism is a triangle raised up, so its volume is the triangle's area times the height. A cylinder is a circle raised up, so its volume is the circle's area times the height. Once you picture a solid as a base lifted through a height, you can find the volume of any of them, and the base area is just one of the flat-shape area formulas you already know.

Why it counts in cubic units

Picture the prism packed tight with little 1 by 1 by 1 unit cubes. One layer across the base holds length × width of them, and there are as many layers as the height. So the total is length × width × height cubes, and that count is the volume. Because three lengths are multiplied, the unit is cubed, the three-dimensional version of the unit squares that measure a flat area.

Working without a unit

This tool does not ask for a unit, and reports the answer as unit³. That keeps it general: enter your three measurements in any one unit, all the same, and read the result in that unit cubed. Centimetres in give cubic centimetres out, metres in give cubic metres out.

A worked example | 4 by 7 by 13

Say the prism is 4 by 7 by 13.

1. Area of the base: 4 × 7 = 28.
2. Times the height: 28 × 13 = 364.

So the volume is 364 unit³. Multiplying all three at once, 4 × 7 × 13, gives the same 364, which is exactly the base area carried up through the height.

The cube, the box, and the prism family

A rectangular prism, a box and a cuboid are three names for the same shape. Make all the edges equal and it becomes a cube; for the same shape framed for everyday capacity, see the volume of box calculator. Step the base from a rectangle to a triangle and you have the triangular prism, another member of the same base-times-height family. And for the skin around this shape rather than the space inside, see the surface area of a rectangular prism calculator.

Questions people ask

What is the volume of a 4 by 7 by 13 rectangular prism?

It is 364 cubic units. Multiply the three dimensions: 4 × 7 × 13, or take the base area 4 × 7 = 28 and multiply by the height 13.

What does "base area times height" mean?

It means working out the area of the base first (length × width for a rectangular prism), then multiplying by the height. The same rule works for any prism and for a cylinder.

Is a rectangular prism the same as a box or a cuboid?

Yes. All three names describe the same six-faced shape, with the same length × width × height volume.

Why is the answer in cubic units?

Because volume multiplies three lengths together, so the unit is cubed. The tool writes it as unit³ since it does not ask which unit you used.

Does base times height work for other shapes?

Yes, for any prism and for cylinders. The volume is always the area of the base times the height, only the base's area formula changes.

References

A note on the principle behind it. The volume of any prism, and of a cylinder, is the area of its base times its height, written V = Bh. The deeper reason the height alone is what matters, even for a slanted solid, is Cavalieri's principle: two solids with the same cross-sectional area at every level have the same volume, set out by Bonaventura Cavalieri in the seventeenth century. For further reading, see Cuboid.

1. The prism volume principle, V = Bh, the area of the base times the height, common to all prisms and cylinders.
2. Bonaventura Cavalieri (17th century), Cavalieri's principle, that solids with equal cross-sections at every height have equal volume.