Cuboid Volume Calculator
Enter the length, width, and height of a cuboid (rectangular box) to get its volume, total surface area, lateral surface area, base area, space diagonal, and all three face diagonals. Switch between metric and imperial units, and see the full step-by-step working below your result.
Formula
Worked example
A box 10 cm long, 6 cm wide, and 4 cm tall: V = 10 x 6 x 4 = 240 cm³; A = 2(60 + 40 + 24) = 248 cm²; space diagonal = sqrt(100 + 36 + 16) = sqrt(152) approx 12.33 cm.
What is a cuboid?
A cuboid (also called a rectangular prism or rectangular box) is a three-dimensional solid with six rectangular faces, twelve edges, and eight vertices. Every pair of opposite faces is parallel and congruent. When all three edge lengths are equal, the cuboid becomes a special case called a cube. Everyday examples include cardboard boxes, shipping containers, rooms, bricks, and books. Because of its flat faces and right-angle corners, the cuboid is one of the simplest solids to measure and work with.
Volume and surface area formulas
Volume is the amount of space a cuboid encloses. It equals length multiplied by width multiplied by height (V = a x b x c). The unit is always cubic, so centimetres give cm³, inches give in³, and so on. Total surface area is the combined area of all six faces: A = 2(ab + ac + bc). Lateral surface area covers only the four side faces and excludes the top and bottom: A_lat = 2c(a + b). The space diagonal is the longest straight line that fits inside the box, stretching from one corner to the opposite corner: d = sqrt(a² + b² + c²). There are also three face diagonals, one for each pair of opposite faces: d_ab = sqrt(a² + b²), d_ac = sqrt(a² + c²), and d_bc = sqrt(b² + c²).
How to use this calculator
Enter the length, width, and height of your cuboid in the same unit. Choose metric (centimetres) or imperial (inches) from the unit selector at the top. All outputs update instantly: volume, total and lateral surface area, base area, space diagonal, and all three face diagonals. Use the "Show your work" panel to follow the step-by-step arithmetic with your exact numbers. If you need the answer in different units, convert your input first (for example, enter dimensions in metres and the volume will be in m³) or multiply the result by the appropriate conversion factor.
Practical uses: shipping, storage, and construction
Volume tells you how much a container holds or how much material fills a space. Divide the volume of a room by the volume of one tile, brick, or bag of concrete to estimate quantities needed. Surface area tells you how much paint, wrapping paper, or sheeting is required to cover the outside. The space diagonal is the critical measurement when you need to know whether a long object (a rod, pipe, or piece of lumber) can fit inside a box when angled from corner to corner. The surface-area-to-volume ratio shown in the insight panel is useful for packaging design: a lower ratio means less material per unit of enclosed space.
Standard cuboid formulas
| Property | Formula | Variables |
|---|---|---|
| Volume | V = a × b × c | a = length, b = width, c = height |
| Total surface area | A = 2(ab + ac + bc) | Sum of all 6 face areas |
| Base area | A_base = a × b | Area of one base (top or bottom) |
| Lateral surface area | A_lat = 2c(a + b) | Four side faces only |
| Space diagonal | d = √(a² + b² + c²) | Interior corner-to-corner |
| Face diagonal a-b | d_ab = √(a² + b²) | Diagonal of the base face |
| Face diagonal a-c | d_ac = √(a² + c²) | Diagonal of the front/back face |
| Face diagonal b-c | d_bc = √(b² + c²) | Diagonal of the left/right face |
All measurements use the same linear unit (cm, in, m, ft, etc.).
Frequently asked questions
What is the difference between a cuboid and a rectangular prism?
They are the same solid. "Rectangular prism" is the standard term in North American textbooks, while "cuboid" is more common in British and international curricula. Both describe a box-shaped solid with six rectangular faces, twelve edges, and eight right-angle vertices.
How do I find the volume of a cuboid?
Multiply the three edge lengths together: V = length x width x height. All three must be in the same unit before you multiply. The result is in cubic units (cm³, m³, in³, ft³, and so on).
What is the space diagonal of a cuboid?
The space diagonal is the straight line connecting two opposite corners of the box, passing through the interior. Its length is sqrt(a² + b² + c²), where a, b, and c are the three edge lengths. It is the longest distance between any two points on or inside the cuboid, and it tells you the maximum length of an object that can fit inside when tilted at the optimal angle.
What is the difference between total surface area and lateral surface area?
Total surface area (A = 2(ab + ac + bc)) includes all six rectangular faces. Lateral surface area (A_lat = 2c(a + b)) includes only the four side faces and leaves out the top and bottom. Lateral area is the relevant figure when you are painting walls but not the floor and ceiling, or wrapping the sides of a package without covering the ends.
Is a cube a special case of a cuboid?
Yes. A cube is a cuboid in which all three edge lengths are equal. It has the same formulas: volume = a³, surface area = 6a², and space diagonal = a x sqrt(3). A cube also has the smallest surface area of any rectangular box that holds the same volume, which is why cube-shaped containers are often the most material-efficient.
How do I convert the volume to litres or gallons?
If your dimensions are in centimetres, divide the volume in cm³ by 1,000 to get litres (1 L = 1,000 cm³). For cubic inches, multiply by 0.016387 to get litres, or by 0.004329 to get US gallons. For cubic feet, multiply by 28.317 to get litres, or by 7.481 to get US gallons.