Cube Calculator - Volume, Area & Diagonal

Calculate the volume, surface area, face diagonal, and space diagonal of a cube from its edge length.

Enter Edge Length

Result

Volume
125
m³
Edge Length (a) 5 m
Volume (V = a³) 125 m³
Surface Area (SA = 6a²) 150 m²
Face Diagonal (d = a√2) 7.071068 m
Space Diagonal (D = a√3) 8.660254 m
Perimeter (P = 12a) 60 m

Step-by-Step Solution

V = a³ = 5³ = 125 m³

Complete Cube Calculator Guide

A cube (also called a regular hexahedron) is one of the five Platonic solids. It is the most familiar three-dimensional shape, featuring six congruent square faces, twelve equal edges, and eight vertices. This calculator computes all essential geometric properties of a cube from a single measurement: the edge length.

Key Cube Formulas

Volume

The space enclosed within the cube. Found by cubing the edge length.

V = a³

Surface Area

Total area of all 6 square faces combined.

SA = 6a²

Face Diagonal

The diagonal across one square face, from corner to corner.

d = a√2

Space Diagonal

The diagonal through the interior, connecting opposite vertices.

D = a√3

Total Edge Length

Sum of all 12 edges of the cube.

P = 12a

Inscribed Sphere Radius

Radius of the largest sphere fitting inside the cube.

r = a / 2

Understanding Face Diagonal vs. Space Diagonal

A cube has two types of diagonals. The face diagonal connects two opposite corners of a single face. Since each face is a square with side length a, the face diagonal is found using the Pythagorean theorem: d = √(a² + a²) = a√2.

The space diagonal (also called the body diagonal) passes through the interior of the cube, connecting two vertices that are not on the same face. It is the longest straight line that can be drawn within the cube. Using the three-dimensional Pythagorean theorem: D = √(a² + a² + a²) = a√3.

Deriving the Volume Formula

The volume of a cube is the amount of three-dimensional space it occupies. Since a cube is a special case of a rectangular prism where all three dimensions are equal, V = length × width × height = a × a × a = a³. This is why raising a number to the third power is called "cubing" it.

Relationships Between Properties

  • Volume from Surface Area: V = (SA/6)3/2
  • Edge from Volume: a = V1/3 (the cube root of the volume)
  • Edge from Surface Area: a = √(SA/6)
  • Edge from Space Diagonal: a = D / √3
  • Edge from Face Diagonal: a = d / √2

The Cube in Mathematics and Nature

The cube holds a special place in mathematics as one of the five Platonic solids -- the only convex polyhedra with congruent regular polygon faces. Cubes tile three-dimensional space perfectly, making them fundamental in crystallography, architecture, and materials science. Common table salt (NaCl) crystals are cubic, and many minerals form cubic crystal systems. In game design and computer graphics, cubes (voxels) are the building blocks for creating complex 3D environments.

Practical Applications

Understanding cube measurements is essential in shipping and logistics (box dimensions), construction (concrete volumes), storage planning (room capacity), and manufacturing (material estimation). The relationships between edge length, volume, and surface area help engineers optimize designs for minimal material usage while meeting volume requirements.