Volume of a cube calculator

The volume of a cube is V = s³, the side length multiplied by itself three times. A cube is a box whose length, width and height are all equal, so l × w × h becomes s × s × s. For a side of 5 cm that is 5 × 5 × 5 = 125 cm³. This is where "cubed" comes from: raising a number to the third power is finding the volume of a cube with that side.

Picture it as counting. A 5 cm cube can be filled with 1 cm cubes in layers: 5 rows of 5 in each layer, 25 per layer, and 5 layers, so 125 small cubes. Volume is that count, and the unit is the small cube, a cubic centimeter here. A cubic centimeter is a milliliter, so the cube holds 125 mL, or 0.125 liters.

The common mistake is squaring instead of cubing. 5² = 25 is the area of one face in square centimeters, not the volume. The surface area of the whole cube is six of those faces, 6s² = 150 cm², which is what you would need to wrap or paint it. Volume is cubic units, area is square units, and a cube happens to have a volume and surface area that are equal only at a side of exactly 6.

Unit conversions cube the conversion factor, and that is the second mistake. A meter is 100 centimeters, so a cubic meter is 100 × 100 × 100 = 1,000,000 cm³, not 100. A yard is 3 feet, so a cubic yard is 27 cubic feet, which is how concrete and mulch are sold. A cubic foot is 1,728 cubic inches, about 7.48 US gallons. The 10 cm cube is the anchor worth remembering: it holds exactly one liter.

Cubing also means the volume grows fast. Double the side to 10 cm and the volume goes from 125 to 1,000 cm³, eight times as much, while the surface goes up only four times. Going the other way needs a cube root: a cube holding 125 cm³ has a side of ∛125 = 5. The ancient problem of doubling the cube asks for the side of a cube with twice the volume; for this one that is ∛250, about 6.300 cm. The Greeks tried to construct that length with a compass and straightedge, and in 1837 it was proved impossible.

Two diagonals are often asked for alongside the volume. Across one face it is s√2, about 7.071 cm here. Through the middle of the cube from one corner to the opposite corner it is s√3, about 8.660 cm, the longest straight object that fits inside. If the sides are not all equal, the solid is a rectangular box, V = l × w × h, which is the box shape on this calculator.