Surface Area of a Cube Calculator
Enter any one measurement of a cube and this calculator works out all the others instantly. Give it a side length, a face diagonal, a space diagonal, the total surface area, or the volume, and it returns the surface area, volume, and both diagonals with a step-by-step breakdown of the math. Switch between metric and imperial units as needed.
Formula
Worked example
A cube with side length 5 cm: face area = 5² = 25 cm², total surface area = 6 x 25 = 150 cm², volume = 5³ = 125 cm³, face diagonal = 5 x sqrt(2) = 7.071 cm, space diagonal = 5 x sqrt(3) = 8.660 cm.
Surface area of a cube formula
A cube has six identical square faces. The area of each face is a² where a is the side length. Multiplying by six gives the total surface area: SA = 6a². This is the amount of flat space that covers the outside of the cube, relevant whenever you need to know how much material, paint, or wrapping covers its exterior. The formula works in any unit: centimetres, inches, metres, or millimetres; just keep your unit consistent throughout.
Solving from any known measurement
You do not always know the side length directly. This calculator accepts any one of five measurements and derives everything else from it. If you have the face diagonal (the line across one square face), divide by sqrt(2) to find the side: a = f / sqrt(2). If you have the space diagonal (the line running through the interior from one corner to the opposite corner), divide by sqrt(3): a = d / sqrt(3). If you know the total surface area, a = sqrt(SA / 6). If you know the volume, a = V^(1/3). Once the side length is found, the rest follows from the standard formulas.
Face diagonal vs space diagonal
A cube has two kinds of diagonals. The face diagonal lies entirely on one square face and connects opposite corners of that face: f = a x sqrt(2), roughly 1.414 times the side. The space diagonal cuts through the hollow interior of the cube from one vertex to the diagonally opposite vertex: d = a x sqrt(3), roughly 1.732 times the side. The space diagonal is always the longest straight line that fits inside a cube of that size, so it sets the minimum clearance needed to pass a rod or pipe through the solid diagonally.
Surface area versus volume and what the ratio means
As a cube grows, its volume increases as a³ while its surface area grows as 6a². The ratio of surface area to volume (6/a) decreases as the cube gets larger, which has practical consequences. A small ice cube has more surface area relative to its mass than a large block and therefore melts faster. Cells in biology must maintain a minimum surface-area-to-volume ratio to exchange nutrients and waste with their environment efficiently. In engineering, small heat exchangers and catalytic surfaces exploit high ratios to maximise heat or chemical transfer per unit volume.
Surface area and volume for common cube sizes
| Side (a) | Surface Area (SA) | Volume (V) | Face Diagonal (f) | Space Diagonal (d) |
|---|---|---|---|---|
| 1 | 6 | 1 | 1.414 | 1.732 |
| 2 | 24 | 8 | 2.828 | 3.464 |
| 3 | 54 | 27 | 4.243 | 5.196 |
| 4 | 96 | 64 | 5.657 | 6.928 |
| 5 | 150 | 125 | 7.071 | 8.660 |
| 6 | 216 | 216 | 8.485 | 10.392 |
| 7 | 294 | 343 | 9.899 | 12.124 |
| 8 | 384 | 512 | 11.314 | 13.856 |
| 9 | 486 | 729 | 12.728 | 15.588 |
| 10 | 600 | 1000 | 14.142 | 17.321 |
All values computed with SA = 6a² and V = a³.
Frequently asked questions
What is the surface area of a cube formula?
The total surface area of a cube is SA = 6a², where a is the side length. A cube has six identical square faces, each with area a², so multiplying by six gives the total exterior area. For example, a cube with a 5 cm side has a surface area of 6 x 25 = 150 cm².
How do I find the surface area if I only know the volume?
First find the side length from the volume: a = V^(1/3). Then apply the surface area formula: SA = 6a². For example, a cube with volume 125 cm³ has a side of 125^(1/3) = 5 cm and a surface area of 6 x 25 = 150 cm². This calculator performs that two-step process automatically when you select "Volume" from the "I know the" dropdown.
What is the difference between the face diagonal and the space diagonal of a cube?
The face diagonal (f) runs across one square face from corner to corner and equals a x sqrt(2). The space diagonal (d) runs through the interior of the cube from one vertex to the opposite vertex and equals a x sqrt(3). For a 5 cm cube, the face diagonal is about 7.07 cm and the space diagonal is about 8.66 cm. The space diagonal is the longest object that fits inside the cube.
Can I calculate the side length from the surface area?
Yes. Rearrange SA = 6a² to get a = sqrt(SA / 6). For a surface area of 150 cm², the side length is sqrt(150 / 6) = sqrt(25) = 5 cm. Select "Surface area (SA)" from the dropdown and enter the value to have this calculator work it out for you.
How is surface area different from volume for a cube?
Surface area measures the total flat area covering the outside of the cube (in square units, e.g. cm²). Volume measures the space inside the cube (in cubic units, e.g. cm³). Surface area matters for tasks like painting, wrapping, or calculating heat transfer. Volume matters for tasks like filling, packing, or measuring capacity. The formulas are SA = 6a² and V = a³ respectively.