Volume Of A Pyramid Calculator

What this calculator does

A pyramid has a flat base that rises to a single point at the top. The great pyramids of Egypt, a tent, a hopper, a glass paperweight. This tool takes a rectangular base, described by a length and a width, together with the height, and works out the volume.

Type the three measurements and you have the space inside.

Using the calculator

1. Type the base width and the base length, the two sides of the rectangular base.
2. Type the height, straight up from the base to the apex.
3. Press Calculate.

All three values have to be positive. There is no unit setting, so the result is a plain number in cubic units of whatever unit you used.

The formula | volume = (length × width × height) ÷ 3

The volume of a rectangular-based pyramid is:

volume = (length × width × height) ÷ 3

The base is a rectangle of area length × width, and the pyramid rises from it to a point. A straight box on that same base and height would hold length × width × height. The pyramid narrows to a tip rather than filling to the top, so it holds exactly a third of the box. That is where the divide by three comes from: volume is one third of the base area times the height.

Why a pyramid is a third of a box

The clean way to hold this in mind: a pyramid is exactly one third of the box, the rectangular prism, that shares its base and its height. Three identical pyramids fit together to fill that box completely.

There is a tidy parallel a dimension down. In the flat world, a triangle is half of the rectangle around it. Step up to solids and a pyramid is a third of its prism. The divide by three here is the same kind of step the divide by two takes for a triangle. For the full box, see the volume of box calculator or the rectangular prism calculator.

Any base, the same one third

This tool works with a rectangular base, but the one-third rule is not fussy about the base shape. A pyramid on any base, triangular, square, hexagonal, holds one third of the base area times the height. Only the way you work out the base area changes. Round the base off into a circle and you have a cone, which follows the very same rule. See the volume of a cone calculator for that round-based cousin.

The height, and the unitless answer

Two things worth being clear on. The height is the straight vertical distance from the base up to the apex, not the slanted face that runs up a side. And this tool does not ask for a unit, so it reports a bare number: work in whatever unit you like, keep all three the same, and read the answer in that unit cubed.

A worked example | 6 by 9 base, height 10

Say the base is 6 wide and 9 long, and the height is 10.

1. Area of the base: 6 × 9 = 54.
2. Times the height: 54 × 10 = 540.
3. Divide by 3: 540 ÷ 3 = 180.

So the volume is 180 cubic units. The middle figure, 540, is the volume of the box on that same base and height, and the pyramid is precisely a third of it.

Questions people ask

What is the volume of a pyramid with a 6 by 9 base and height 10?

It is 180 cubic units. Find the base area 6 × 9 = 54, multiply by the height 10, and divide by 3.

Why do you divide by 3?

Because a pyramid is one third of the box with the same base and height. Three of them fill that box.

Is the height the slanted face?

No. It is the straight vertical distance from the base to the apex. The slanted face is the slant height, which is used for surface area.

Does it work for non-rectangular bases?

The rule does. Any pyramid's volume is one third of its base area times its height. This tool just computes the base area as length × width for a rectangle.

What unit is the answer in?

Whatever unit you used, cubed. The tool reports a plain number, so if you worked in centimetres the answer is in cubic centimetres.

References

A note on where this comes from. That a pyramid is exactly one third of the prism sharing its base and height is an old result. Euclid proved it in his Elements, Book XII, Proposition 7, by cutting a triangular prism into three pyramids of equal volume. The same fact also drops out of Cavalieri's principle, that two solids with the same cross-sectional area at every level have the same volume. And it mirrors the way a triangle is half of its surrounding rectangle down in the plane. For further reading, see Pyramid (geometry).

1. Euclid, Elements, Book XII, Proposition 7 (c. 300 BC), dividing a triangular prism into three equal pyramids, the classical proof that a pyramid is one third of its prism.
2. Cavalieri's principle, underpinning a pyramid being one third of the prism that shares its base and height.
3. The planar parallel: a triangle is one half of the rectangle around it, as a pyramid is one third of its prism.