Cube Calculator: Find Volume, Surface Area, and Diagonal
Enter any one property of a cube (side length, volume, surface area, face diagonal, or space diagonal) and this calculator solves all the others instantly. Choose your unit, then switch the "Solve for" mode to match whichever measurement you already know. The step-by-step panel shows exactly how each result is reached.
Formula
Worked example
A cube with side 5 cm: volume = 5^3 = 125 cm^3; surface area = 6 x 25 = 150 cm^2; face diagonal = 5 x sqrt(2) = 7.071 cm; space diagonal = 5 x sqrt(3) = 8.660 cm.
What is a cube and what properties does it have?
A cube is the simplest three-dimensional solid: six identical square faces, twelve equal edges, and eight vertices. Because every edge has the same length, a single measurement (the side length, usually called a) completely determines the cube. From that one number you can calculate the volume, the total surface area, the diagonal across any face, and the longest diagonal through the interior. This calculator works in all five directions - enter whichever property you know and it solves the rest.
The four cube formulas explained
Volume (V = a^3): the cube multiplied three times gives the three-dimensional space enclosed. A 5 cm cube holds 125 cm^3. Surface area (S = 6a^2): the cube has six square faces, each of area a^2, so the total paint area is six times that. Face diagonal (f = a x sqrt(2)): the diagonal across one square face is found by the Pythagorean theorem applied to a right triangle with two legs of length a, giving sqrt(a^2 + a^2) = a x sqrt(2). Space diagonal (d = a x sqrt(3)): the longest line inside the cube connects two opposite corners. By the 3-D version of the Pythagorean theorem: d = sqrt(a^2 + a^2 + a^2) = a x sqrt(3).
Reverse-solving: find the side from any property
If you know the volume, take its cube root: a = V^(1/3). If you know the surface area, use a = sqrt(S / 6). For the face diagonal: a = f / sqrt(2). For the space diagonal: a = d / sqrt(3). The "I know the" dropdown at the top switches the active input field so only the value you actually have is visible, keeping the form clean.
Practical applications
Cube geometry appears in packaging design (how much cardboard does a cubic box need?), construction (how many cubic metres of concrete fill a square form?), manufacturing (volume and material use of cubic components), and science (unit cells in crystallography are often cubic). The volume-to-surface-area ratio - which equals a / 6 - tells you how efficiently a cube encloses space: larger cubes have a higher ratio, meaning less surface per unit of volume, which is why bulk shipping containers are cost-effective.
Cube formulas at a glance
| Property | Symbol | Formula |
|---|---|---|
| Side length | a | a (given) |
| Volume | V | a^3 |
| Surface area | S | 6 a^2 |
| Face diagonal | f | a x sqrt(2) |
| Space diagonal | d | a x sqrt(3) |
| Inverse: side from V | a | V^(1/3) |
| Inverse: side from S | a | sqrt(S / 6) |
| Inverse: side from f | a | f / sqrt(2) |
| Inverse: side from d | a | d / sqrt(3) |
All properties of a cube expressed in terms of side length a.
Frequently asked questions
How do I find the volume of a cube?
Multiply the side length by itself three times: V = a x a x a = a^3. For a cube with side 4 cm, the volume is 4^3 = 64 cm^3. If you only know the surface area, divide by 6, take the square root to get the side, then cube it.
What is the difference between face diagonal and space diagonal?
The face diagonal connects two opposite corners of one square face and stays on the surface of the cube. Its length is a x sqrt(2). The space diagonal connects two opposite corners of the entire cube and passes through the interior. Its length is a x sqrt(3), making it the longest straight line you can fit inside the cube.
How do I find the side length if I only know the volume?
Take the cube root of the volume: a = V^(1/3). For example, if V = 27 m^3 then a = 27^(1/3) = 3 m. In this calculator, select "Volume (V)" from the dropdown, enter your volume, and the side length appears automatically.
How do I calculate surface area from the space diagonal?
First recover the side: a = d / sqrt(3). Then compute surface area: S = 6a^2 = 6 x (d / sqrt(3))^2 = 6d^2 / 3 = 2d^2. So the surface area is simply twice the square of the space diagonal.
Why does a cube have the minimum surface area among rectangular boxes of equal volume?
Among all rectangular boxes (cuboids) with a fixed volume, the one with equal side lengths - that is, a cube - minimises the surface area. This is a classical result from the isoperimetric inequality in three dimensions. It is why hexagonal honeycombs and cubic unit cells appear in nature: they pack space with the least material.
Does this calculator support imperial units?
Yes. Use the "Length unit" selector to switch between millimetres, centimetres, metres, kilometres, inches, feet, yards, and miles. All five outputs (side, volume, surface area, face diagonal, space diagonal) update immediately in the chosen unit.