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diagonal of a square — Definition, Formula & Examples

The diagonal of a square is the straight line segment that connects two opposite corners (vertices). Every square has exactly two diagonals, and they are always equal in length.

A diagonal of a square with side length ss is a line segment joining two non-adjacent vertices. Its length equals s2s\sqrt{2}, derived by applying the Pythagorean theorem to the right triangle formed by two sides and the diagonal.

Key Formula

d=s2d = s\sqrt{2}
Where:
  • dd = Length of the diagonal
  • ss = Length of one side of the square

How It Works

A diagonal splits the square into two congruent right triangles. Each triangle has legs equal to the side length ss and a hypotenuse equal to the diagonal dd. By the Pythagorean theorem, d2=s2+s2=2s2d^2 = s^2 + s^2 = 2s^2, so d=s2d = s\sqrt{2}. The two diagonals of a square bisect each other at right angles, meeting at the center of the square.

Worked Example

Problem: Find the length of the diagonal of a square with side length 10 cm.
Write the formula: Use the diagonal formula for a square.
d=s2d = s\sqrt{2}
Substitute: Plug in the side length.
d=102d = 10\sqrt{2}
Evaluate: Compute the decimal value.
d≈14.14 cmd \approx 14.14 \text{ cm}
Answer: The diagonal is 102≈14.1410\sqrt{2} \approx 14.14 cm.

Why It Matters

Knowing the diagonal helps you find distances across square rooms, screens, or tiles without measuring corner to corner. It also appears in coordinate geometry when you calculate the distance between opposite vertices of a square on a grid.

Common Mistakes

Mistake: Doubling the side length instead of multiplying by √2.
Correction: The diagonal is not twice the side. It equals s2s\sqrt{2} (about 1.414 times the side), which comes from the Pythagorean theorem, not from doubling.

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