Last updated: 2026-06-10

What a cube is

A cube is a three-dimensional solid bounded by six identical square faces meeting at right angles, with all twelve edges equal in length. Because every dimension is the same, a single measurement — the side length s — fully determines the cube, and from it follow the volume, surface area, face diagonal, space diagonal, and total edge length. A cube is the special case of a rectangular prism in which length, width, and height are all equal.

The formulas

For a cube with side length s:

Property	Formula	Meaning
Volume	V = s³	Space enclosed inside the cube
Surface area	A = 6s²	Total area of all six faces
Face diagonal	$d_f = s\sqrt{2}$	Diagonal across one square face
Space diagonal	$d_s = s\sqrt{3}$	Corner-to-corner through the interior
Total edge length	L = 12s	Sum of all 12 edges

Surface area

A cube has 6 identical square faces, each with area s². The total surface area is simply 6 × s². For s = 10 cm: A = 6 × 100 = 600 cm².

Deriving the face diagonal

Each face is a square with side s. The diagonal of a square is found by the Pythagorean theorem:

$d_f = \sqrt{s^2 + s^2} = \sqrt{2s^2} = s\sqrt{2}$

For s = 10 cm: $d_f = 10\sqrt{2} \approx 14.14$ cm.

Deriving the space diagonal

The space diagonal stretches from one corner of the cube to the diametrically opposite corner. Apply the 3D Pythagorean theorem — first find the face diagonal d_f, then extend to the opposite vertex:

$d_s = \sqrt{d_f^2 + s^2} = \sqrt{2s^2 + s^2} = \sqrt{3s^2} = s\sqrt{3}$

For s = 10 cm: $d_s = 10\sqrt{3} \approx 17.32$ cm. The space diagonal is always longer than the face diagonal ($\sqrt{3} > \sqrt{2}$).

Relation to the rectangular prism

A cube is simply a rectangular prism where l = w = h = s. Substituting into the rectangular prism formulas:

• V = l·w·h → s·s·s = s³
• A = 2(lw + lh + wh) → 2(s² + s² + s²) = 6s²
• Space diagonal = $\sqrt{l^2 + w^2 + h^2} \to \sqrt{3s^2} = s\sqrt{3}$

Cubic scaling

Because volume scales as s³, a change in side length has a disproportionately larger effect on volume than on surface area:

Side factor	Volume factor	Surface area factor
×2	×8	×4
×3	×27	×9
×½	×⅛	×¼

A shipping cube of 20 cm holds 8× as much as one of 10 cm, even though it is only twice as wide.

Worked examples

Example 1 — Ice cube

A standard ice cube has s ≈ 2.5 cm (0.025 m).

Volume: V = 0.025³ = 0.000015625 m³ = 15.625 cm³ ≈ 15.6 mL
Surface area: A = 6 × 0.025² = 0.00375 m² = 37.5 cm²
Space diagonal: $d_s = 0.025 \times \sqrt{3} \approx 0.0433$ m ≈ 4.33 cm

Example 2 — Shipping box

A cubic shipping carton with s = 30 cm:

Volume: V = 30³ = 27,000 cm³ = 27 litres
Surface area: A = 6 × 900 = 5,400 cm² = 0.54 m² of cardboard
Total edge length: L = 12 × 30 = 360 cm = 3.6 m of tape needed for all edges

Example 3 — Finding the side from the volume

When the volume is known to be 1,000 cm³ (= 0.001 m³), the side length is found by taking the cube root:

s = ∛V = ∛1000 = 10 cm

For V = 2 m³: s = ∛2 ≈ 1.26 m.

Frequently Asked Questions (FAQ)

How do I find a cube's space diagonal?

The space diagonal of a cube with side s is d = s√3. It runs from one corner through the center to the opposite corner. For example, a cube with s = 10 cm has a space diagonal of 10√3 ≈ 17.32 cm.

What's the difference between a face diagonal and a space diagonal?

A face diagonal lies entirely on one face of the cube and has length s√2. A space diagonal cuts through the interior from one vertex to the opposite vertex and has length s√3. For s = 10 cm: face diagonal ≈ 14.14 cm, space diagonal ≈ 17.32 cm.

How do I find the side length from the volume of a cube?

Rearrange V = s³ to get s = ∛V. For example, if V = 1000 cm³ (= 0.001 m³), then s = ∛1000 = 10 cm. For V = 1 m³, s = ∛1 = 1 m.

Why is the surface area of a cube 6s²?

A cube has 6 identical square faces, each with area s². The total surface area is therefore 6 × s². For s = 5 cm, A = 6 × 25 = 150 cm².