Cube Calculator: Volume, Surface Area & Diagonals
A cube is defined by one edge length because all twelve edges are equal. From that single measurement, you can determine its volume, total surface area, face diagonal, and space diagonal. The calculator above can also work backwards from any one of those values.
What is a cube?
A cube is a three-dimensional solid made from six identical square faces. It has 12 equal edges and 8 vertices, with three mutually perpendicular edges meeting at every vertex. A cube is therefore a special case of a rectangular prism in which length, width, and height are equal.
Cube formulas
The core formulas follow directly from the cube's six equal square faces and three equal dimensions. The diagonal formulas come from applying the Pythagorean theorem first to a face and then through the solid.
| Known value | Find the edge | Then calculate |
|---|---|---|
| Edge a | a | V, S, f, d |
| Volume V | ∛V | S, f, d |
| Surface area S | √(S/6) | V, f, d |
| Face diagonal f | f/√2 | V, S, d |
| Space diagonal d | d/√3 | V, S, f |
How the diagonals work
The face diagonal is the diagonal of one square face. Pythagoras gives f² = a² + a², so f = a√2. For the space diagonal, apply the theorem once more using the face diagonal and the remaining edge: d² = f² + a² = 3a², which gives d = a√3.
Volume and surface area
Cube volume is the product of its three equal dimensions: a × a × a = a³. Surface area is the sum of six equal square faces, each with area a², giving 6a².
V = a³ = 4³ = 64 cm³S = 6a² = 6 × 4² = 96 cm²f = a√2 = 4√2 ≈ 5.657 cmd = a√3 = 4√3 ≈ 6.928 cmWorked example from volume
Suppose a cubic container has a volume of 125 cm³, but its edge length is unknown. Take the cube root first; once the edge is known, the remaining properties follow from the standard formulas.
a = ∛125 = 5 cmS = 6 × 5² = 150 cm²f = 5√2 ≈ 7.071 cmd = 5√3 ≈ 8.660 cmUnits matter
The numerical formulas stay the same when the unit changes, but the resulting unit changes with the dimension. An edge measured in centimeters produces surface area in square centimeters and volume in cubic centimeters.
| Measurement | Formula | If a is in cm |
|---|---|---|
| Edge | a | cm |
| Face diagonal | a√2 | cm |
| Space diagonal | a√3 | cm |
| Surface area | 6a² | cm² |
| Volume | a³ | cm³ |
Cube properties at a glance
FAQ
Use V = a³, where a is the edge length.
Take the cube root: a = ∛V.
The face diagonal crosses one square face. The space diagonal passes through the interior from one vertex to the opposite vertex.
Yes. Rearrange S = 6a² to get a = √(S/6), then calculate the other properties.