What is volume?
Volume is the amount of space enclosed by a three-dimensional shape, measured in cubic units. If the dimensions are in centimetres, the volume is in cubic centimetres (cm^3, or millilitres); if they are in metres, the volume is in cubic metres (m^3). Volume is to 3D what area is to 2D and what length is to 1D, and the units always carry a power that matches the number of dimensions.
Two unit reminders that come up often: 1 cubic metre = 1,000 litres = 1,000,000 cm^3, and 1 cubic foot is about 7.481 US gallons. We will use these conversions in the examples and again at the end of the guide.
Volume always scales with the cube of length. If you double every dimension of a shape, the volume goes up by a factor of 2^3 = 8. This is why a kitten weighs almost nothing compared to a fully grown lion of the same general shape: bigger animals have disproportionately more volume (and therefore mass) than smaller ones.
Cube and rectangular prism (box)
Cube: V = s^3, where s is the side length. A cube with side 5 cm has volume 5^3 = 125 cm^3, or 125 millilitres.
Rectangular prism (box): V = length * width * height. A box 10 by 5 by 4 cm has volume 10 * 5 * 4 = 200 cm^3.
Boxes are the most common volume calculation in daily life: shipping boxes, swimming pools, room dimensions, gardening beds. Always use the same units for all three dimensions, and the result comes out in those units cubed.
Cylinder
Cylinder: V = pi * r^2 * h, where r is the radius of the circular base and h is the height. The first part, pi * r^2, is the area of the circle; multiplying by the height stacks the circle up into a column.
Example: a cylindrical water tank with radius 30 cm and height 100 cm has volume V = 3.14159 * 900 * 100 = about 282,743 cm^3, or about 283 litres.
Cylinders show up everywhere: water tanks, food cans, pipes (the volume is then the cross-section area times the length), engine cylinders. Just remember it is pi r squared times height, not pi r times height squared.
Sphere
Sphere: V = (4/3) * pi * r^3. Notice the cube of the radius: as with all volumes, doubling the radius multiplies the volume by 8.
Example: a basketball has a radius of about 12 cm, so its volume is (4/3) * 3.14159 * 12^3 = (4/3) * 3.14159 * 1728 = about 7,238 cm^3, or about 7.2 litres of air.
The formula comes from calculus (or, historically, from Archimedes' clever proof comparing a sphere to a cylinder), and it is worth memorising: it appears in physics, astronomy, and engineering whenever something can be approximated as round.
Cone and pyramid
Cone: V = (1/3) * pi * r^2 * h. A cone has exactly one third the volume of a cylinder with the same base and height.
Pyramid (rectangular base): V = (1/3) * length * width * height. A square pyramid is V = (1/3) * s^2 * h.
The shared factor of 1/3 is not a coincidence. Both cones and pyramids are shapes that taper from a base to a single point (the apex), and any such shape has one third the volume of the prism or cylinder with the same base and height. This is one of the most elegant results in geometry.
Example: an ice-cream cone with radius 3 cm and height 10 cm holds V = (1/3) * 3.14159 * 9 * 10 = about 94.2 cm^3, or 94 millilitres. A tetrahedron-shaped tent with a 4 m by 4 m square base and 3 m height has V = (1/3) * 16 * 3 = 16 m^3.
Quick reference: every formula in one place
Use this as a cheat sheet. All formulas assume consistent units; the result is in those units cubed.
- Cube: V = s^3
- Rectangular prism: V = l * w * h
- Cylinder: V = pi * r^2 * h
- Sphere: V = (4/3) * pi * r^3
- Cone: V = (1/3) * pi * r^2 * h
- Pyramid (rectangular base): V = (1/3) * l * w * h
- Square pyramid: V = (1/3) * s^2 * h
- Triangular prism: V = (1/2 * base * height of triangle) * length
Volume of an irregular shape
Most real objects are not perfect cubes or spheres. There are two practical strategies for finding the volume of an irregular shape.
Decomposition. Break the shape into simple pieces (boxes, cylinders, cones), compute each volume, and add them up. An L-shaped room becomes two rectangular prisms; a glass with a tapered bottom becomes a cylinder plus a cone frustum. This is how architects estimate concrete and how carpenters work out timber needs.
Displacement. Submerge the object in a container of water and measure how much the water level rises. The volume of the object equals the volume of displaced water (1 mL of water = 1 cm^3). This was Archimedes' famous insight when he stepped into a full bath. Displacement is the only reliable way to measure the volume of an oddly shaped, solid, waterproof object.
Volume vs surface area
Volume measures the space inside a shape; surface area measures the outside. They scale differently: if you double every dimension, volume goes up by 8 and surface area only by 4. This is why small animals lose heat fast (their surface-area-to-volume ratio is large) and why large animals must work harder to cool down.
When you are buying paint, wrapping a present, or estimating heat transfer, you want surface area, not volume. When you are buying concrete, filling a tank, or computing mass from density, you want volume. The two are partners but they answer different questions; always double-check which one your problem actually needs.
Unit conversions you will need
Common cubic-unit conversions worth memorising:
- 1 m^3 = 1,000 litres = 1,000,000 cm^3
- 1 cm^3 = 1 millilitre
- 1 cubic foot = 7.481 US gallons = 28.317 litres
- 1 cubic yard = 27 cubic feet = 764.6 litres
- 1 US gallon = 3.785 litres = 231 cubic inches