Diagonal of a Square Calculator

Calculate the diagonal of a square from side length, area, or perimeter. Learn the diagonal formula (d = side × √2) with step-by-step examples and explanations.

Diagonal of a Square Calculator

Calculate square diagonal from side, area, or perimeter

Formula: diagonal = side × √2 ≈ side × 1.4142

Understanding Square Diagonals

What is a Diagonal of a Square?

A diagonal is a line segment that connects two non-adjacent vertices (corners) of a square. Every square has two diagonals with special properties:

  • Equal length: Both diagonals are always the same length
  • Perpendicular: They intersect at right angles (90°)
  • Bisecting: Each diagonal cuts the other in half
  • Angle bisectors: They divide the square's 90° corners into two 45° angles

The Diagonal Formula

The relationship between a square's side and its diagonal comes from the Pythagorean theorem:

d = a × √2

where d is the diagonal and a is the side length

Why this works: A diagonal divides the square into two congruent right triangles (45-45-90 triangles). The diagonal is the hypotenuse, and the two sides are the legs.

Using the Pythagorean theorem: a² + a² = d²

Solving for d: d = √(2a²) = a√2

Worked Examples

Example 1: Finding Diagonal from Side

A square has a side length of 10 inches. What is its diagonal?

Solution:

d = 10 × √2 = 10 × 1.4142 = 14.14 inches

Example 2: Finding Side from Diagonal

A square has a diagonal of 20 cm. What is the side length?

Solution:

side = 20 ÷ √2 = 20 ÷ 1.4142 = 14.14 cm

Example 3: Finding Diagonal from Area

A square has an area of 64 ft². What is its diagonal?

Solution:

First find the side: side = √64 = 8 ft

Then find diagonal: d = 8 × √2 = 11.31 ft

Or use the direct formula: d = √(2 × 64) = √128 = 11.31 ft

Quick Reference Values

Side Length Diagonal Area
1 unit 1.414 units 1 unit²
5 units 7.071 units 25 units²
10 units 14.142 units 100 units²
12 units 16.971 units 144 units²
20 units 28.284 units 400 units²

Frequently Asked Questions

What is the diagonal of a square of side 1?

The diagonal is √2 (approximately 1.4142 units). This is a fundamental relationship: when the side equals 1, the diagonal equals the square root of 2.

What is the side of a square of diagonal 1?

The side length is √2/2 or approximately 0.707 units. This is calculated as: side = 1 ÷ √2 = √2/2 ≈ 0.707

How do I find the side of a square given the diagonal?

Divide the diagonal length by √2 (approximately 1.4142):

side = diagonal ÷ √2

For example, if the diagonal is 10 units: side = 10 ÷ 1.4142 ≈ 7.07 units

Why is the diagonal always √2 times the side?

This comes from the Pythagorean theorem. When a diagonal splits a square into two right triangles:

  • Both legs of the triangle equal the side length (a)
  • The hypotenuse equals the diagonal (d)
  • By Pythagorean theorem: a² + a² = d²
  • Simplifying: 2a² = d², so d = a√2

Can a square have different diagonal lengths?

No! Both diagonals of a square are always equal in length. This is one of the defining properties of a square. If the diagonals are different lengths, the shape is not a square (it might be a rectangle or other quadrilateral).

What's the relationship between diagonal and area?

You can find the area directly from the diagonal using:

Area = d² ÷ 2

And conversely: diagonal = √(2 × area)

For example, if diagonal = 10 units, then area = 100 ÷ 2 = 50 square units

How does this relate to 45-45-90 triangles?

A square's diagonal creates two 45-45-90 triangles (isosceles right triangles). In these special triangles:

  • The two legs are equal (the square's sides)
  • The hypotenuse (diagonal) = leg × √2
  • Each acute angle is 45°

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