Cube Formulas – Complete Reference

A cube is a three-dimensional solid with 6 equal square faces, 12 equal edges, and 8 vertices (corners). All sides are equal in length (s). The cube is to 3D what the square is to 2D — the most regular and symmetric solid shape.

Volume V = s³ (side cubed)
Surface Area SA = 6s² (6 square faces)
Face Diagonal = s × √2 ≈ 1.414s
Space Diagonal = s × √3 ≈ 1.732s
Side from Volume: s = ∛V

Reference Table

Side (s)	Volume (s³)	Surface Area (6s²)	Space Diagonal (s√3)
1	1	6	1.732
2	8	24	3.464
3	27	54	5.196
4	64	96	6.928
5	125	150	8.660
10	1000	600	17.321

Worked Examples

Example 1 – Rubik's Cube

A standard Rubik's Cube has sides of 5.7 cm. Find its volume and surface area.

V = 5.7³ = 185.19 cm³
SA = 6 × 5.7² = 6 × 32.49 = 194.94 cm²

Example 2 – Shipping Carton

A cubic shipping carton has sides of 40 cm. How much cardboard is needed (surface area)?

SA = 6 × 40² = 6 × 1600 = 9600 cm² = 0.96 m²

Example 3 – Find Side from Volume

A cubic room has volume 216 m³. What is the side length and how tall is the ceiling?

s = ∛216 = 6 m (side = ceiling height)
SA = 6 × 36 = 216 m² (total wall/floor/ceiling area)

Why Two Types of Diagonals?

A cube has two types of diagonals:

Face diagonal: Diagonal across one square face = s√2. Connects two vertices on the same face.
Space diagonal: Diagonal through the interior of the cube = s√3. Connects two opposite vertices (the longest internal distance).

Face diagonal = √(s²+s²) = s√2
Space diagonal = √(s²+s²+s²) = s√3

Real-World Applications

Packaging: Cubic boxes maximize volume while minimizing surface area compared to other rectangular boxes
Ice cubes: Volume determines how much cooling a cube provides
Sugar cubes: Standard sugar cubes are approximately 1 cm³
Dice: Standard dice are cubes — each face has equal probability
Architecture: Cubic building designs, modular housing units
Computing: 3D voxels (volumetric pixels) used in 3D modelling and CT scans are tiny cubes

Volume of a Cube Calculator