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diagonal of a cube — Definition, Formula & Examples

The diagonal of a cube (also called the space diagonal) is the straight line segment that connects two opposite corners of the cube, passing through its interior. For a cube with side length ss, this diagonal has length s3s\sqrt{3}.

Given a cube with edge length ss, the space diagonal is the segment joining two vertices that share no common face. Its length is derived by applying the Pythagorean theorem twice: first across a face diagonal (s2s\sqrt{2}), then from that diagonal up through the third dimension, yielding s2+s2+s2=s3\sqrt{s^2 + s^2 + s^2} = s\sqrt{3}.

Key Formula

d=s3d = s\sqrt{3}
Where:
  • dd = Length of the space diagonal of the cube
  • ss = Length of one edge of the cube

How It Works

A cube has four space diagonals, and they all have the same length. To find one, you extend the 2D diagonal idea into three dimensions. First, compute the face diagonal across the base: s2s\sqrt{2}. Then treat that face diagonal and the vertical edge as legs of a right triangle. The hypotenuse of that triangle is the space diagonal: (s2)2+s2=s3\sqrt{(s\sqrt{2})^2 + s^2} = s\sqrt{3}.

Worked Example

Problem: Find the space diagonal of a cube with side length 4 cm.
Write the formula: Use the space diagonal formula.
d=s3d = s\sqrt{3}
Substitute: Plug in s=4s = 4.
d=43d = 4\sqrt{3}
Evaluate: Compute the decimal approximation.
d≈4×1.732=6.928 cmd \approx 4 \times 1.732 = 6.928 \text{ cm}
Answer: The space diagonal is 43≈6.934\sqrt{3} \approx 6.93 cm.

Why It Matters

The space diagonal determines the longest straight object that fits inside a box-shaped room or container. In 3D modeling and physics, it also gives the distance between opposite corners of any cubic cell, which is essential for calculating atomic radii in crystal structures (body-centered cubic lattices).

Common Mistakes

Mistake: Using s2s\sqrt{2} instead of s3s\sqrt{3}.
Correction: s2s\sqrt{2} is the face diagonal (2D, across one face). The space diagonal passes through the interior of the cube and requires accounting for all three dimensions, giving s3s\sqrt{3}.

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