Volume of Cubes and Cuboids: Formulas with Examples

Volume is the amount of three-dimensional space a solid occupies — how much water a box can hold, or how much grain fits in a container. For a cube, every edge has the same length a, so its volume is simply a multiplied by itself three times. A cuboid has three different dimensions — length l, breadth b and height h — and its volume is the product of all three. Both formulas are just the base area multiplied by the height.

Volume matters everywhere liquids and loose materials are stored or moved. Water tanks, shipping cartons, oil drums and grain silos are all sized by volume. In school problems, volume questions often hide inside everyday stories: how many litres a tank holds, how many small cubes fit inside a big box, or how much concrete fills a foundation pit.

Key Concepts

• Edge (a): the side length of a cube.
• Length, breadth, height (l, b, h): the three dimensions of a cuboid.
• Volume: measured in cubic units (cm³, m³). A cube 1 cm on each side holds 1 cm³.
• Capacity: volume expressed in litres or millilitres; 1 m³ = 1000 litres and 1 cm³ = 1 ml.
• All three dimensions must be in the same unit before multiplying.

Formulas

• Volume of a cube: a³ — the edge length cubed.
• Volume of a cuboid: l × b × h — length times breadth times height.
• Capacity conversion: divide cm³ by 1000 to get litres, since 1000 cm³ = 1 litre.
• These assume solid, right-angled boxes with no missing parts.

Worked Example

Find the volume of a cuboid with l = 12 cm, b = 7 cm and h = 5 cm. Also find the volume of a cube of edge 9 cm.

1. Cuboid: V = l × b × h = 12 × 7 × 5.
2. Compute 12 × 7 = 84; then 84 × 5 = 420. The cuboid's volume is 420 cm³.
3. Cube: V = a³ = 9³ = 9 × 9 × 9 = 81 × 9 = 729. The cube's volume is 729 cm³.
4. In litres, the cuboid holds 420 ÷ 1000 = 0.42 litres.
5. Bonus: a tank 2 m × 1.5 m × 1 m holds 3 m³ = 3000 litres of water.

Common Misconceptions

• Adding the dimensions instead of multiplying: volume is l × b × h, not l + b + h. Adding gives a length, not a volume.
• Writing the answer in cm²: volume is three-dimensional, so the unit must be cubic (cm³, m³).
• Mixing units: if the length is in metres and the breadth in centimetres, convert to one unit before multiplying, or the answer will be wrong by a large factor.

Real-World Applications

• Sizing water tanks, swimming pools and fuel containers by capacity.
• Calculating how many cartons fit in a shipping container.
• Estimating concrete needed for foundations and pillars.
• Packing and storage planning in warehouses and homes.

A cube's volume is a³ and a cuboid's is l × b × h — each is the base area times the height. Multiply all three dimensions in the same unit, report the answer in cubic units, and divide cm³ by 1000 whenever the question asks for litres.