Volume Formulas: Boxes, Cylinders, Cones and Spheres

Boxes and cubes: length × width × height

Volume counts how many unit cubes fit inside a solid. A rectangular box measuring 3 cm by 4 cm by 5 cm holds 60 cubic centimeters. That is the same as its base area, 3 × 4 = 12 square centimeters, multiplied by its height of 5 cm. Base area times height is the idea that carries through to cylinders and, with one adjustment, to cones.

A cube is a box whose edges are all equal, so its volume is the edge length cubed. A cube with 10 cm edges holds 1,000 cubic centimeters, which is exactly one liter. A cube with 1 m edges holds one cubic meter, or 1,000 liters.

Scaling behaves more dramatically for volume than for area. If every length of a solid grows by a factor k, the volume grows by k³. Doubling the edges makes a box eight times larger, and even a 1.5 times increase gives 3.375 times the volume. This is why a container that looks only a little bigger can hold a lot more.

• Box: length × width × height
• Cube: edge³
• Scale lengths by k and volume scales by k³

Cylinders, cones and spheres

A cylinder is a circle extruded upward, so its volume is the circle's area times the height: πr²h. A cooking pot 20 cm across (radius 10 cm) and 10 cm deep holds π × 100 × 10, about 3,141.6 cubic centimeters, or about 3.14 liters. Treat a drinks can as a cylinder with a 3.3 cm radius and a 12.2 cm height and you get about 417 cubic centimeters, more than its labeled 350 mL because of the headspace and the tapered top.

A cone holds exactly one third of the cylinder with the same base and height: πr²h ÷ 3. An ice cream cone with a 2.5 cm radius and a 10 cm height holds about 65.45 cubic centimeters. Pyramids follow the same rule: base area × height ÷ 3.

A sphere's volume is (4/3)πr³. A ball with a 10 cm radius holds about 4,188.79 cubic centimeters, roughly 4.19 liters. A size 5 soccer ball, about 22 cm in diameter, comes to about 5,575 cubic centimeters. Archimedes showed that a sphere fills exactly two thirds of the cylinder that just encloses it, one with radius r and height 2r, and you can check that the formulas agree.

• Cylinder: πr²h
• Cone: πr²h ÷ 3
• Sphere: (4/3)πr³
• Sphere surface area: 4πr²

Why cones and pyramids are divided by three

A pointed solid gets narrower as you move up. Its cross-sections shrink with the square of the distance from the tip, so the stack of slices adds up to much less than a straight column. With calculus, a cone or pyramid with base area S and height h has a cross-section of S × (x/h)² at height x from the tip; adding those slices from 0 to h gives S × h ÷ 3.

There is also a picture you can hold in your head. Draw lines from the center of a cube to its eight corners and the cube splits into six identical square pyramids, each with a face of the cube as its base and half the edge as its height. If the edge is a, the six pyramids together make a³, so each is a³ ÷ 6. The prism with the same base and height would be a² × a/2 = a³ ÷ 2, and a³ ÷ 6 is exactly one third of that.

A classroom demonstration makes the point physically: fill a cone with water and pour it into a cylinder with the same base and height, and it takes three cones to fill the cylinder. Forgetting the divide-by-three is the most common error with these shapes, and it makes the answer three times too big.

From cubic units to liters and gallons

Capacity is the volume measured on the inside of a container, so always use internal dimensions. The key metric equivalences are 1 cubic centimeter = 1 milliliter, 1,000 cubic centimeters = 1 liter, and 1 cubic meter = 1,000 liters. A liter is simply a cube 10 cm on each side.

Before you multiply, make sure every length is in the same unit. A tank listed as 30 cm by 40 cm by 0.5 m needs the 0.5 m turned into 50 cm first, giving 30 × 40 × 50 = 60,000 cubic centimeters, or 60 liters.

Gallons need extra care because there are two of them. The US liquid gallon is defined as exactly 231 cubic inches, which is 3.785411784 liters. The UK imperial gallon is 4.54609 liters, about 20 percent larger. A cubic foot is 28.316846592 liters, or about 7.48 US gallons. Recipes, fuel economy figures and aquarium sizes from different countries can quietly mean different gallons.

• 1 cubic centimeter = 1 mL
• 1 liter = 1,000 cubic centimeters
• 1 cubic meter = 1,000 liters
• 1 US gallon = 231 cubic inches = 3.785411784 liters
• 1 imperial gallon = 4.54609 liters
• 1 cubic foot = 28.316846592 liters, about 7.48 US gallons

Worked examples: tanks, tubs, pools and garden beds

A common 20-gallon long aquarium measures about 30 by 12 by 12 inches on the outside. That is 4,320 cubic inches, and dividing by 231 gives about 18.7 US gallons, or about 70.8 liters. The nominal size is rounded, the glass takes up some of the space, and you never fill to the brim, so the real amount of water is lower still once gravel and decorations go in.

A bathtub with internal dimensions of 120 cm by 70 cm, filled to 40 cm, would hold 120 × 70 × 40 = 336,000 cubic centimeters, or 336 liters, if it were a perfect box. Most tubs slope and have rounded corners, so treat the box calculation as an upper bound.

A round above-ground pool 12 feet across with water 4 feet deep is a cylinder with a 6 ft radius: π × 36 × 4 is about 452.4 cubic feet, roughly 3,384 US gallons or about 12,810 liters. That figure is what you would use to estimate how much water to order or how much treatment chemical a label calls for.

For a raised garden bed 8 feet long, 4 feet wide and 1 foot deep, the volume is 32 cubic feet. Bulk soil is often sold by the cubic yard, which is 3 × 3 × 3 = 27 cubic feet, so the bed needs about 1.19 cubic yards. Ordering a little extra makes sense because soil settles after watering.

• 30 × 12 × 12 in tank: 4,320 cubic inches, about 18.7 US gallons
• 120 × 70 × 40 cm tub as a box: 336 liters (upper bound)
• 12 ft round pool, 4 ft deep: about 3,384 US gallons
• 8 × 4 × 1 ft garden bed: 32 cubic feet, about 1.19 cubic yards