How To Calculate Volume From Dimensions

If you want to calculate the volume of a three-dimensional figure, you need to know the shape of the figure. To calculate the volume from the dimensions of some figures, you have to use calculus, but for many regular figures, the application of geometry produces a simple formula. Remember that all the dimensions you use in any given calculation must be in the same units.

The easiest shape for which to calculate volume is a rectangular container, such as a fish tank or a show box. It has three sides of lengths ​a​, ​b​ and ​c​. You probably already know that you can calculate the area of a cross section of the box by multiplying its length, ​a​, by its width, ​b​. Now extend this area by the depth, ​c​, and you have the volume:

The volume of a rectangle with sides a, b and c is:

\(V_{rect}=a\times b\times c\)

A cube is a special kind of rectangle that has all three sides of equal length, ​a​.

The volume of a cube is:

\(V_{cube}=a\times a\times a=a^3\)

A cylindrical container, such as a pill container, has a circular cross section and a certain length (​h​). You can measure both of these with a ruler. The diameter of the circle (​d​) is easier to measure than the radius (​r​), but the formula works best with the radius, so just convert using the formula ​r​ = ​d​/2. The area of the circular cross section is then π​r​² or π​d​²/ 4. Extend that area along the length (​h​) of the cylinder to get the volume:

\(V_{cylinder}=\pi \times r^2\times h = \pi \times \frac{d^2}{4}\times h\)

If you measure from one side of the widest part of a sphere to the opposite side, you get the diameter, and half of this is the radius (​r​). You can calculate the area of the circle at the sphere's widest point using the area formula π​r​², but extrapolating to volume isn't simple and requires integral calculus. Fortunately, you don't have to do this yourself, because it's already been figured out:

\(V_{sphere}=\frac{4}{3}\times \pi \times r^3\)

An ellipsoid is an elongated sphere. To calculate its volume, first locate the center and measure the lengths of the three perpendicular axes ​a​, ​b​ and ​c​ from that point to the surface of the ellipsoid. You can now calculate its volume:

\(V_{ellipsoid}=\frac{4}{3}\times \pi \times a\times b\times c\)

The shape of the base of a pyramid can be any polygon,, and there is a single general formula that allows to to calculate the volume of it:

\(V_{pyramid}=\frac{1}{3}\times A_b\times h\)

where ​A​_b is the area of the base and ​h​ is the height.

If the pyramid has a triangular base, visualize tipping the base on one end. It's a triangle with base ​b​ and height ​l​. You calculate the area using the formula (1/2) × ​b​ × ​l​, so the volume of the pyramid is:

\(V_{tri-pyr}=\frac{1}{6}\times b\times l\times h\)

If the pyramid has a rectangular base of length ​l​ and width ​w​, the area of the base is ​l​ × ​w​. The volume of the pyramid is then:

\(V_{rect-pyr}=\frac{1}{3}\times l\times w\times h\)

A cone is a shape with a circular cross-section that tapers to a point. If the radius of the cone at its widest point is ​r​ and the length of the cone ​h​, you can find the volume using calculus, or you can do as most people do and look it up.

\(V_{cone}=\frac{1}{3}\times \pi\times r^2\times h\)