A cube is a three-dimensional solid with six identical square faces, twelve equal edges, and eight vertices — and every measurement of a cube follows from a single number: the edge length $a$.
Definition: A cube is a regular hexahedron — all six faces are congruent squares, all edges equal, all angles 90°. Symbol: Edge length = $a$ Volume formula: $V = a^3$ Total Surface Area formula: $\text{TSA} = 6a^2$ Lateral Surface Area formula: $\text{LSA} = 4a^2$ Face diagonal formula: $d_f = a\sqrt{2}$ Space diagonal formula: $d_s = a\sqrt{3}$ Type: 3D geometric solid Used in: Geometry, mensuration, engineering, crystallography
Full Definition
A cube is a special case of a rectangular prism (cuboid) in which all three dimensions — length, width, and height — are equal. Because all edges have the same length $a$, every face is a square with side $a$, and every formula reduces to a power of $a$.
A cube has 6 faces, 12 edges, and 8 vertices. By Euler's formula for polyhedra, $F - E + V = 6 - 12 + 8 = 2$, which holds.
Why The Cube Formula Matters
The cube appears in physics as the unit cell of the simplest crystal lattice structure (simple cubic), in chemistry as the model for ionic solid arrangements, and in everyday measurement as the basis of the cubic unit of volume ($\text{cm}^3$, $\text{m}^3$). When René Descartes (1596–1650, France) formalised coordinate geometry, the cube became the natural 3D extension of the square — and its volume formula $a^3$ is why we call the third power of a number its "cube."
Variable Key
Symbol | Meaning | Unit |
|---|---|---|
$a$ | Edge length (all edges equal in a cube) | length (cm, m, etc.) |
$V$ | Volume | cubic units (cm³, m³) |
$\text{TSA}$ | Total Surface Area (all 6 faces) | square units (cm², m²) |
$\text{LSA}$ | Lateral Surface Area (4 side faces only) | square units (cm², m²) |
$d_f$ | Face diagonal (diagonal across one square face) | length |
$d_s$ | Space diagonal (diagonal through the interior of the cube) | length |
Cube Formula Derivation
Volume: A cube is a stack of $a$ layers, each layer being a square of side $a$ with area $a^2$. Total volume = $a^2 \times a = a^3$.
Total Surface Area: The cube has 6 identical square faces, each with area $a^2$. TSA $= 6 \times a^2 = 6a^2$.
Lateral Surface Area: The 4 side faces (excluding top and bottom) each have area $a^2$. LSA $= 4a^2$.
Face diagonal: Using the Pythagorean theorem on one square face with sides $a$ and $a$: $$d_f = \sqrt{a^2 + a^2} = \sqrt{2a^2} = a\sqrt{2}$$
Space diagonal: Applying the Pythagorean theorem in 3D — the space diagonal is the hypotenuse of a right triangle with legs $a$ (one edge) and $d_f = a\sqrt{2}$ (a face diagonal): $$d_s = \sqrt{a^2 + (a\sqrt{2})^2} = \sqrt{a^2 + 2a^2} = \sqrt{3a^2} = a\sqrt{3}$$
Worked Examples of Cube Formula
Example 1: Volume and TSA of a cube
A cube has edge length $a = 5$ cm. Find the volume and total surface area.
$$V = a^3 = 5^3 = 125 \text{ cm}^3$$
$$\text{TSA} = 6a^2 = 6 \times 5^2 = 6 \times 25 = 150 \text{ cm}^2$$
Final answer: Volume = 125 cm³, Total Surface Area = 150 cm²
Example 2: Finding edge length from volume
A cube has volume 343 cm³. Find the edge length.
$$a^3 = 343$$ $$a = \sqrt[3]{343} = 7 \text{ cm}$$
Final answer: Edge length = 7 cm
Common Confusions With The Cube Formula
Cube vs cuboid: A cube is a special cuboid where all three dimensions are equal ($l = w = h = a$). A cuboid's volume is $l \times w \times h$; for a cube this simplifies to $a^3$. Always check whether all three dimensions are equal before applying the cube formula.
TSA vs LSA: Total Surface Area includes all 6 faces ($6a^2$); Lateral Surface Area includes only the 4 side faces ($4a^2$). Problems about painting just the sides, or wrapping without a lid, use LSA.
Face diagonal vs space diagonal: The face diagonal ($a\sqrt{2}$) lies on the surface; the space diagonal ($a\sqrt{3}$) passes through the interior. They are different lengths and answer different geometric questions.
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