Cube Formula – Volume, Surface Area & Diagonal
Cube Formula – Volume, Surface Area & Diagonal
TL;DR
This guide covers every cube formula you need — volume, total surface area, lateral surface area, face diagonal, and space diagonal — each derived from first principles and illustrated with a worked example. After reading you will be able to solve any standard exam question on cubes without looking up a formula sheet.
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 = a³
Total Surface Area formula: TSA = 6a²
Lateral Surface Area formula: LSA = 4a²
Face diagonal formula: d_f = a√2
Space diagonal formula: d_s = a√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 (cm³, m³). When René Descartes (1596–1650, France) formalised coordinate geometry, the cube became the natural 3D extension of the square — and its volume formula a³ 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) |
| TSA | Total Surface Area (all 6 faces) | square units (cm², m²) |
| 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 a². Total volume = a² × a = a³.
Total Surface Area: The cube has 6 identical square faces, each with area a². TSA = 6 × a² = 6a².
Lateral Surface Area: The 4 side faces (excluding top and bottom) each have area a². LSA = 4a².
Face diagonal: Using the Pythagorean theorem on one square face with sides a and a: d_f = √(a² + a²) = √(2a²) = a√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 face diagonal): d_s = √(a² + (a√2)²) = √(a² + 2a²) = √(3a²) = a√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 = a³ = 5³ = 125 cm³
TSA = 6a² = 6 × 5² = 6 × 25 = 150 cm²
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.
a³ = 343 → a = ∛343 = 7 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 × w × h; for a cube this simplifies to a³. Always check whether all three dimensions are equal before applying the cube formula.
TSA vs LSA: Total Surface Area includes all 6 faces (6a²); Lateral Surface Area includes only the 4 side faces (4a²). Problems about painting just the sides, or wrapping without a lid, use LSA.
Face diagonal vs space diagonal: The face diagonal (a√2) lies on the surface; the space diagonal (a√3) passes through the interior. They are different lengths and answer different geometric questions.
Frequently Asked Questions
What is the cube formula for volume?
The cube formula for volume is V = a³, where a is the length of one edge. Because all edges of a cube are equal, the volume is simply the edge length multiplied by itself three times.
How is the space diagonal of a cube found?
Apply the Pythagorean theorem twice. First find the face diagonal: d_f = a√2. Then use that as one leg and edge a as the other: d_s = √(a² + 2a²) = a√3.
What is the difference between a cube and a square?
A square is a 2D shape with four equal sides. A cube is the 3D solid built by extending a square into depth — six square faces enclosing a volume. Area of a square face is a²; volume of the cube is a³.
If the volume of a cube is doubled, what happens to the edge length?
The edge length scales by 23≈1.26. Doubling the volume does not double the edge length - it scales it by the cube root of 2.