A cube is the most symmetric of all three-dimensional shapes: six identical square faces, twelve equal edges and eight vertices. Because every measurement derives from the single edge length a, knowing any one property lets you recover every other. This calculator accepts the edge length, the volume, the face area, or the space diagonal as input and returns all six standard cube properties — volume, total surface area, face area, face diagonal and space diagonal — together with a worked step and an interactive SVG diagram.
How it works
Select what you already know from the “I know the …” drop-down, enter the value, and every result updates instantly. Under the hood the tool inverts the relevant formula:
| Known | Formula used to find edge a |
|---|---|
| Edge a | already known |
| Volume V | a = V^(1/3) (cube root) |
| Face area A | a = √A |
| Space diagonal d | a = d / √3 |
Once the edge is resolved, all other properties follow from exact closed-form expressions with no approximations beyond floating-point precision.
The formulas
For a cube with edge length a:
- Volume: V = a³
- Face area (one face): A = a²
- Total surface area (six faces): SA = 6a²
- Face diagonal (corner to corner across one face): fd = a · √2
- Space diagonal (corner to corner through the centre): d = a · √3
The face diagonal follows from Pythagoras applied to one square face: fd² = a² + a² = 2a². The space diagonal extends that into three dimensions: d² = a² + a² + a² = 3a², so d = a·√3.
Worked example
Suppose you have a wooden cube whose volume is 125 cm³. What are all its properties?
- Edge: a = 125^(1/3) = 5 cm
- Face area: 5² = 25 cm²
- Total surface area: 6 × 25 = 150 cm²
- Face diagonal: 5 × √2 ≈ 7.071 cm
- Space diagonal: 5 × √3 ≈ 8.660 cm
These are the five numbers the calculator shows simultaneously the moment you type 125 in “Volume” mode — no manual steps needed.
Common reference values
| Edge a | Volume a³ | Surface area 6a² | Space diagonal a·√3 |
|---|---|---|---|
| 1 | 1 | 6 | 1.732 |
| 5 | 125 | 150 | 8.660 |
| 10 | 1 000 | 600 | 17.321 |
| 100 | 1 000 000 | 60 000 | 173.205 |
Doubling the edge multiplies the volume by 8 and the surface area by 4 — a consequence of cubic and quadratic scaling respectively.
Formula notes
The cube is a special case of the rectangular cuboid where all three dimensions are equal. It is also a regular hexahedron — one of the five Platonic solids. The equality of all edges means a single variable fully determines the entire geometry, making calculations straightforward and the shape uniquely efficient: among all cuboids of a given surface area, the cube encloses the maximum volume (the isoperimetric theorem for polyhedra).
All arithmetic runs entirely in your browser using JavaScript’s IEEE 754 double-precision floating-point — accurate to about 15 significant digits. No data is sent anywhere.