Cube Surface Area and Volume Calculator
This calculator works out the surface area, volume, face diagonal and space diagonal of a cube from a single measurement, the length of one side. A cube is a three-dimensional shape with six identical square faces, twelve equal edges and eight corners, so knowing just the side length lets you derive every other dimension using standard geometry formulas. Enter the side length and choose a unit, millimetres, centimetres, metres, kilometres, inches, feet or yards, and the calculator instantly returns the surface area, the combined area of all six faces, the volume, how much space the cube occupies, the face diagonal, the line running across one square face, and the space diagonal, the line running through the cube from one corner to the opposite corner. Full working is shown underneath, setting out each formula and the exact numbers used, so you can see how every figure was reached rather than just the final answer. This is useful for jobs such as working out how much material is needed to cover a cube-shaped object, checking how much a container will hold, or confirming whether a cube-shaped item will fit through a gap using its space diagonal. Results are calculated from exact geometric formulas and displayed rounded to three decimal places, so they stay accurate for whichever unit of length you choose to work in.
Cube Dimensions
About a Cube
A cube is a three-dimensional solid with six equal square faces, twelve equal edges, and eight vertices. All sides are equal length. It is a special case of a rectangular box (cuboid) where length = width = height = s.
Key formulas:
Surface area: SA = 6s²
Volume: V = s³
Face diagonal: d₁ = s√2
Space diagonal: d₃ = s√3
Full Working
How to Calculate the Surface Area of a Cube
A cube has six identical square faces. Each face has an area of s² (side length squared). The total surface area is the sum of all six faces:
SA = 6s²
For example, a cube with side length 5 cm has a surface area of 6 × 5² = 6 × 25 = 150 cm². Surface area is measured in square units (cm², m², in², and so on).
How to Calculate the Volume of a Cube
Volume measures how much three-dimensional space the cube occupies. Because all three dimensions of a cube are equal, volume is simply the side length raised to the power of three:
V = s³
A cube with side length 5 cm has a volume of 5 × 5 × 5 = 125 cm³. Volume is measured in cubic units (cm³, m³, in³, and so on).
Diagonals of a Cube
There are two types of diagonal in a cube:
- Face diagonal: runs across a single square face from one corner to the opposite corner. By Pythagoras, the face diagonal equals the hypotenuse of a right triangle with both legs equal to s, so the face diagonal = s√2. For a 5 cm cube: 5 × 1.41421 = 7.071 cm.
- Space diagonal (body diagonal): runs from one corner of the cube to the diagonally opposite corner, passing through the interior. It equals s√3. You can derive this by applying Pythagoras twice: first to find the face diagonal (s√2), then treating that as one leg and s as the other to find the space diagonal: √((s√2)² + s²) = √(2s² + s²) = √(3s²) = s√3. For a 5 cm cube: 5 × 1.73205 = 8.66 cm.
Worked Example
A cube has a side length of 5 cm.
| Property | Formula | Calculation | Result |
|---|---|---|---|
| Surface area | 6s² | 6 × 5² = 6 × 25 | 150 cm² |
| Volume | s³ | 5 × 5 × 5 | 125 cm³ |
| Face diagonal | s√2 | 5 × 1.41421 | 7.071 cm |
| Space diagonal | s√3 | 5 × 1.73205 | 8.66 cm |
Common Applications
Cube calculations appear in many practical contexts: estimating how much paint is needed to coat a box (surface area), calculating the capacity of a storage container (volume), determining whether a cube-shaped object will fit through a doorway (space diagonal), and in engineering when specifying cubic components.
Related Calculators
- Maths and Stats Calculators: full list of geometry and statistics tools.
- Volume Calculator: volume for multiple solid shapes.
- Surface Area Calculator: surface area for multiple solid shapes.
- Area of a Triangle Calculator: triangle area using base and height.
Method: Standard Euclidean geometry formulas for a regular hexahedron (cube). Surface area SA = 6s²; Volume V = s³; Face diagonal d₁ = s√2; Space diagonal d₃ = s√3. No external source required: these are universally accepted geometric identities.
This calculator is for educational purposes. Results are exact within floating-point precision. All formulas assume a perfect cube (all sides equal). Displayed values are rounded to three decimal places for surface area and volume, and three decimal places for diagonals.