Cube Calculator: Volume, Surface Area, Diagonals & More
A cube is one of the simplest three-dimensional solids, yet a single edge length determines almost every measurement you usually need. Enter the edge, volume, surface area, face diagonal, or space diagonal above and the calculator derives the rest.
What is a cube?
A cube is a three-dimensional solid bounded by six identical square faces. Three faces meet at each vertex, and adjacent edges are perpendicular. It can be viewed as a special rectangular prism in which length = width = height = a. This is why one measurement is enough to determine its volume, surface area, and diagonals.
Cube formulas
These four relationships are the core of the calculator. The first two describe the amount of space and exposed surface; the last two describe straight-line distances inside the cube.
V = a³S = 6a²f = a√2d = a√3| Measurement | Formula | Dimension | What it represents |
|---|---|---|---|
| Edge | a | length | One of the cube's 12 equal edges |
| Volume | a³ | cubic | Three-dimensional space inside |
| Surface area | 6a² | square | Total area of all six faces |
| Face diagonal | a√2 | length | Corner-to-corner across one face |
| Space diagonal | a√3 | length | Corner-to-corner through the solid |
How to calculate the volume
Start with the familiar prism idea: volume = base area × height. A cube's base is a square, so its base area is a². Because every edge is equal, the height is also a. Therefore:
This also explains why volume grows quickly. If the edge is doubled, each of the three dimensions doubles, so the volume becomes 2 × 2 × 2 = 8 times as large.
How to calculate surface area
Every face is a square with area a², and there are six congruent faces. Add them together:
Surface area grows with the square of the edge. Doubling the edge therefore makes the total surface area 4 times larger, not 2 times larger.
How the diagonals work
There are two different diagonal measurements worth separating. A face diagonal stays on one square face. A space diagonal passes through the cube's interior and connects opposite vertices.
Reverse calculations: when the edge is unknown
You do not always start with the edge. Because the four measurements are mathematically linked, you can rearrange the formulas and recover a first.
| If you know | Recover the edge with | Then calculate |
|---|---|---|
| Volume V | a = ∛V | surface, face diagonal, space diagonal |
| Surface area S | a = √(S/6) | volume, both diagonals |
| Face diagonal f | a = f/√2 | volume, surface, space diagonal |
| Space diagonal d | a = d/√3 | volume, surface, face diagonal |
Worked examples
V = 5³ = 125 cm³S = 6 × 5² = 150 cm²f = 5√2 ≈ 7.071 cmd = 5√3 ≈ 8.660 cma = ∛216 = 6 cmS = 6 × 6² = 216 cm²f = 6√2 ≈ 8.485 cmd = 6√3 ≈ 10.392 cmCube values at a glance
The table below is also a useful mental model for the live charts above. Notice how the volume accelerates faster than surface area as the edge increases.
| Edge a | Volume a³ | Surface 6a² | Face diagonal | Space diagonal |
|---|---|---|---|---|
| 1 | 1 | 6 | 1.414 | 1.732 |
| 2 | 8 | 24 | 2.828 | 3.464 |
| 3 | 27 | 54 | 4.243 | 5.196 |
| 4 | 64 | 96 | 5.657 | 6.928 |
| 5 | 125 | 150 | 7.071 | 8.660 |
| 10 | 1,000 | 600 | 14.142 | 17.321 |
How to read the live charts
The curves are not decorative. Each one is generated from the same equations used by the calculator, with your current input marked on the curve. That means the shape of each graph has a mathematical reason.
V = a³. The curve gets progressively steeper because a appears three times.
S = 6a². It rises faster than a straight line but slower than volume.
f = a√2 and d = a√3. Both are straight relationships with fixed multipliers.
The cube drawing scales with the current edge and labels the face and space diagonals.
Units and common mistakes
| Quantity | Correct unit type | Example when a = 4 cm |
|---|---|---|
| Edge | cm | 4 cm |
| Face / space diagonal | cm | 5.657 cm / 6.928 cm |
| Surface area | cm² | 96 cm² |
| Volume | cm³ | 64 cm³ |
Cube vs. “cubed” number
In algebra, “cubing” a number means raising it to the third power: n³ = n × n × n. A perfect cube is the result when an integer is cubed, such as 3³ = 27 or 5³ = 125. This idea is closely related to cube volume because a cube with integer edge length can be imagined as a block of unit cubes.
FAQ
V = a³, where a is the edge length.
Take the cube root: a = ∛V. For example, ∛125 = 5.
Multiply the area of one square face by six: S = 6a².
The face diagonal crosses one square face. The space diagonal connects opposite vertices through the cube's interior.
The space diagonal is longer: a√3 > a√2 for every positive edge length.
Yes. Rearrange the formula to a = √(S/6), then use that edge to calculate the other properties.
Volume becomes 8 times larger, surface area becomes 4 times larger, and both diagonals become 2 times larger.
Yes. The curves are sampled directly from the cube equations shown on this page; the current input is highlighted on each curve.