Cube Calculator
Find the volume, surface area, face diagonal, and space diagonal of a cube. Enter any one value you know and get all the others instantly.
Quick Examples
What Is a Cube?
A cube is a three-dimensional solid with six identical square faces, twelve edges of equal length, and eight vertices (corners). It is one of the five Platonic solids — the most symmetric 3D shapes possible. Because every edge has the same length, a single measurement determines the entire cube: its volume, surface area, and both diagonals. This cube calculator takes any one of those values and computes the rest.
Cubes are everywhere: dice, sugar cubes, storage boxes, ice cubes, Rubik's cubes, and shipping cartons. Students meet them in geometry, engineers size them for packaging, and builders estimate concrete volumes with them. Whenever you know just one cube measurement, the formulas below recover everything else.
Cube Formulas
Let s be the edge length of the cube:
Where the Diagonal Formulas Come From
Each face of a cube is a square, so the face diagonal follows the square's diagonal: f = s√2 (about 1.414s). The space diagonal runs through the interior from one corner to the opposite. It forms a right triangle with one edge (s) and one face diagonal (s√2), so by the Pythagorean theorem d² = s² + (s√2)² = 3s², giving d = s√3 ≈ 1.732s.
How to Use the Cube Calculator
- Choose which measurement you already know: edge length, volume, surface area, or space diagonal.
- Enter the value (any positive number, including decimals).
- Click Calculate Cube. The tool first recovers the edge length, then derives volume, surface area, and both diagonals.
The calculator solves in both directions — from an edge to everything else, or backward from a volume or diagonal to the edge.
Worked Examples
Example 1: Given the edge length
A storage box is a cube with 3 ft edges. Its volume is 3³ = 27 ft³, its surface area is 6 × 3² = 54 ft², its face diagonal is 3√2 ≈ 4.24 ft, and its space diagonal is 3√3 ≈ 5.20 ft.
Example 2: Given the volume
A cube-shaped tank holds 64 liters. The edge is ∛64 = 4 units, the surface area is 6 × 4² = 96 square units, the face diagonal is 4√2 ≈ 5.66, and the space diagonal is 4√3 ≈ 6.93.
Example 3: Given the surface area
A gift box needs 150 in² of wrapping paper. The edge is √(150 ÷ 6) = √25 = 5 inches, the volume is 5³ = 125 in³, the face diagonal is 5√2 ≈ 7.07 in, and the space diagonal is 5√3 ≈ 8.66 in.
Example 4: Given the space diagonal
A cube's interior diagonal measures 12 cm. The edge is 12 ÷ √3 ≈ 6.93 cm, the volume is about 332.55 cm³, the surface area is about 288 cm², and the face diagonal is about 9.80 cm.
Solving Backward: Every Inversion
Each property determines the edge, so you can always work backward:
| If you know | Edge length s = |
|---|---|
| Volume V | ∛V |
| Surface area SA | √(SA ÷ 6) |
| Space diagonal d | d ÷ √3 |
| Face diagonal f | f ÷ √2 |
Real-Life Applications
- Packaging and shipping: convert a required box volume into the edge length to order the right carton size.
- Construction: estimate concrete for cube-shaped footings from a target volume.
- Cooking: relate a recipe's volume (a 4-inch cube pan holds 64 in³ ≈ 1 liter) to pan dimensions.
- Gaming: dice and voxel geometry are built on exact cube properties.
- Storage: check whether a cube of known diagonal fits through a doorway or into a shelf.
Common Mistakes
- Mixing up area and volume units: surface area is square units, volume is cubic units — 54 ft² of cardboard is not 54 ft³ of space.
- Confusing the two diagonals: the face diagonal (s√2 ≈ 1.414s) is shorter than the space diagonal (s√3 ≈ 1.732s). Use the right one when checking fit.
- Forgetting the 6 faces: surface area is 6s², not s² — a frequent error when estimating paint or wrapping paper.
- Mixing up cube (shape) with cubing (x³): the volume formula uses s³, which is why "multiply by itself twice more" inherited the name — but this calculator is about the geometric solid.
Frequently Asked Questions
How do you find the volume of a cube?
Cube the edge length: V = s³. A cube with edge 3 has volume 27 cubic units.
What is the surface area of a cube?
Six times the square of the edge: SA = 6s², because a cube has six identical square faces. An edge-3 cube has surface area 54 square units.
What is the space diagonal of a cube?
The line running corner to corner through the interior: d = s√3 ≈ 1.732s. An edge-3 cube has a space diagonal of about 5.20.
How do you find the edge length from the volume?
Take the cube root: s = ∛V. A cube with volume 64 has edge length 4.
What is the difference between face diagonal and space diagonal?
The face diagonal runs across one square face (s√2 ≈ 1.414s); the space diagonal passes through the interior (s√3 ≈ 1.732s). The space diagonal is always longer.
How many faces, edges, and vertices does a cube have?
6 square faces, 12 edges, and 8 vertices. All edges are equal and all faces are identical squares.