Volume Formula

Unlike area, which only requires multiplying two dimensions, volume requires multiplying three dimensions (or base area and height). Because different 3D shapes have unique geometry, they each rely on a different volume formula.

Here are the standard formulas for the six most common 3D geometric shapes:

Every one of these multiplies three lengths together, which is why the answer is always in cubic units. Five of the six also fall into two families, which is worth knowing because it lets you handle bases these formulas do not name.

• Prisms and cylinders have the same cross-section all the way along, so V = base area × h. The cube and the rectangular prism are prisms; the cylinder is the circular case.
• Pyramids and cones taper from a flat base to a single apex, so V = (1/3) × base area × h. That covers a pyramid on a triangular or hexagonal base too, once you can work out the base area.

In both families h is the perpendicular height: the straight-line distance between the two parallel faces, or from the base plane up to the apex. It is not the slant height along a sloping face.

The sphere belongs to neither family. It has no flat base and no constant cross-section, so it keeps its own formula, V = (4/3) × π × r³. Solids outside these families, such as a barrel or a torus, need formulas of their own as well.

Area multiplies two lengths and is reported in square units. Volume multiplies three, so the unit is cubed. One cubic foot is the space inside a cube measuring one foot on every edge.

This is also why cubic conversions are not the same as linear ones. There are 12 inches in a foot, but 12 × 12 × 12 = 1,728 cubic inches in a cubic foot. Likewise 3 feet make a yard, but 27 cubic feet make a cubic yard. Converting a volume with a linear factor is one of the fastest ways to be wrong by a factor of several hundred.

The inch, foot, and yard are each defined exactly in metres: 1 inch = 0.0254 m, 1 foot = 0.3048 m, and 1 yard = 0.9144 m. Cubing those definitions gives cubic factors that are exact too, so 1 cubic foot is exactly 0.028316846592 cubic metres rather than a rounded approximation of it.

Volume does not grow in step with length, and this catches people out when they resize something.

• Double one dimension of a box and the volume doubles. Double all three and it grows eightfold, because 2 × 2 × 2 = 8.
• A cube or a sphere has only one independent length, so doubling it always multiplies the volume by eight. A sphere of twice the radius holds eight times as much, not twice as much.
• In a cylinder or cone the radius is squared and the height is not. Doubling the radius quadruples the volume; doubling the height only doubles it. Widening a container is far more effective than heightening it.

Let's calculate the volume for each of the five shapes using real numbers:

• Example 1: Cube
  Find the volume of a box with equal side lengths of 4 cm.
  V = s³ = 4³ = 4 × 4 × 4 = 64 cm³.
• Example 2: Rectangular Prism (Box)
  Find the volume of a package that is 10 inches long, 5 inches wide, and 6 inches high.
  V = l × w × h = 10 × 5 × 6 = 300 in³.
• Example 3: Cylinder
  Find the volume of a soda can with a base radius of 3 cm and a height of 12 cm.
  V = πr²h = π × 3² × 12 = π × 9 × 12 = 108π ≈ 339.29 cm³.
• Example 4: Sphere
  Find the volume of a toy ball with a radius of 6 inches.
  V = (4/3)πr³ = (4/3) × π × 6³ = (4/3) × π × 216 = 288π ≈ 904.78 in³.
• Example 5: Cone
  Find the volume of a funnel with a base radius of 3 inches and a height of 8 inches.
  V = (1/3)πr²h = (1/3) × π × 3² × 8 = (1/3) × π × 9 × 8 = 24π ≈ 75.40 in³.
• Example 6: Pyramid (rectangular base)
  Find the volume of a display case shaped like a pyramid with a base 6 ft long, 4 ft wide, and a height of 9 ft.
  V = (1/3) × l × w × h = (1/3) × 6 × 4 × 9 = (1/3) × 216 = 72 ft³.
  The matching box, with the same base and the same height, would hold 216 ft³, exactly three times as much.

• Forgetting to convert units. If your height is in feet but your width is in inches, you will calculate an incorrect volume. Make sure all dimensions are converted to the same unit before multiplying.
• Squaring instead of cubing. Volume is three-dimensional, meaning the output unit must be cubic (e.g. cm³, yd³), not square (e.g. cm², yd²).
• Using diameter instead of radius. For cylinders, spheres, and cones, the formulas require the radius (r). If you are given a diameter, remember to divide it by 2 first.

When working specifically with conical containers, calculating the volume involves an extra step if you are given the slant height instead of the vertical height. In these cases, you must use the Pythagorean theorem to find the vertical height before solving:

height = √(slant_height² − radius²)

To skip the manual geometry, the cone volume calculator resolves the height and volume directly.

• Euclid, Elements, Book XII, Proposition 10: any cone is a third part of the cylinder with the same base and equal height. The corollary to Proposition 7 gives the same relationship for a pyramid and its prism. These support the one-third factor used above.
• NIST Special Publication 811, Appendix B.9, which lists the inch (2.54 E−02 m), foot (3.048 E−01 m), and yard (9.144 E−01 m) as exact conversions to the metre. These support the exact cubic factors in the unit section.