Cube: worked example
| Side length (s) | 5 |
|---|---|
| Volume | 125 |
| Surface area | 150 |
| Face diagonal | 7.1 |
| Space diagonal | 8.7 |
Cube: formula
Use the common edge length in one unit. A face diagonal lies on a square surface; a space diagonal connects opposite vertices through the solid. Surface area and volume have different dimensions and cannot be substituted when ordering coating or fill.
Cube: changing Side length (s)
| Side length (s) | Volume | Surface area | Face diagonal | Space diagonal |
|---|---|---|---|---|
| 4 | 64 | 96 | 5.7 | 6.9 |
| 4.5 | 91.1 | 121.5 | 6.4 | 7.8 |
| 5 (example) | 125 | 150 | 7.1 | 8.7 |
| 5.5 | 166.4 | 181.5 | 7.8 | 9.5 |
| 6 | 216 | 216 | 8.5 | 10.4 |
Cube: compare input ranges
| Input | Tested from | To | Volume: low input | Volume: high input | Swing |
|---|---|---|---|---|---|
| Side length (s) | 4.5 | 5.5 | 91.1 | 166.4 | 75.3 (60%) |
Cube: individual input comparisons
| Input | Value | Volume | Surface area | Face diagonal | Space diagonal |
|---|---|---|---|---|---|
| Side length (s) | 4 | 64 | 96 | 5.7 | 6.9 |
| 4.5 | 91.1 | 121.5 | 6.4 | 7.8 | |
| 5 | 125 | 150 | 7.1 | 8.7 | |
| 5.5 | 166.4 | 181.5 | 7.8 | 9.5 | |
| 6 | 216 | 216 | 8.5 | 10.4 |
Cube: which assumptions move the result?
Cube: values and definitions
| Quantity and definition | Value used |
|---|---|
| Side length (s) | 5 |
| Volume | 125 |
| Surface area | 150 |
| Face diagonal | 7.1 |
| Space diagonal | 8.7 |
Cube: half and double Side length (s)
| Side length (s) | Volume |
|---|---|
| 2.5 | 15.6 |
| 5 | 125 |
| 10 | 1,000 |
Common error: Doubling a side and expecting double the volume. Doubling every edge multiplies the volume by eight and the surface by four.
Real cubes have thickness, so inside and outside dimensions differ. Rounded corners reduce volume slightly. A box with unequal sides is a rectangular prism, not a cube.
The results scale steeply, so measuring errors in the side are amplified in volume.
Edge measurement and squared or cubed units
Assume a cube's edge is 10 centimeters. Its six faces total 600 cm², and its volume is 1,000 cm³. If the entered edge is instead 10.1 centimeters, the same equations give 612.06 cm² and 1,030.301 cm³.
A one-percent edge overstatement has produced a 2.01% area difference and a 3.0301% volume difference. These are calculated sensitivity figures for the assumed measurement, not a manufacturing tolerance. Unit conversion also follows the power: 10 centimeters is 0.1 meter, so this original cube occupies 0.001 m³ and has 0.06 m² of surface.
Applying only the linear centimeter-to-meter factor to a square or cubic result gives the wrong dimensions.
OpenStax Prealgebra 2e §9.6: cube volume and surface areaGeometry calculators: Equilateral Triangle Calculator, Trigonometry Calculator, Golden Ratio Calculator, Midpoint Calculator, Rectangle Area & Perimeter Calculator, Circle Area & Circumference Calculator.
A cube needs one number, and everything else follows from it. The space diagonal is the longest straight rod that fits inside, which is useful when the question is whether a long object will fit in a box.
Confirm the object really is a cube. A box with three different edges is a rectangular prism, and these results will not fit it.
See the rectangular prism calculator for boxes, the sphere, cylinder and pyramid pages for other solids, and the Pythagorean theorem calculator for diagonals. The exponent calculator shows cubing, and the unit conversion pages change cubic units to liters or gallons.
How does surface area change when the side doubles?
It becomes four times larger, while the volume becomes eight times larger, which is why bigger boxes need proportionally less material per unit of capacity.
What is the volume of a cube?
The side length cubed. A cube with a side of 5 has a volume of 125 cubic units.
How do I find the space diagonal of a cube?
Multiply the side length by the square root of 3, which is about 1.732 times the side.