# Square Diagonal Calculator

Calculate the diagonal of a square from its side length, or find the side from the diagonal. Also shows area and perimeter. Free online square diagonal calculator.

## What this calculates

Calculate the diagonal of a square from the side length, or find the side length from the diagonal. This calculator also shows the area and perimeter of the square.

## Inputs

- **Calculate From** — options: Side Length to Diagonal, Diagonal to Side Length — Choose whether to calculate the diagonal from the side or vice versa.
- **Side Length (s)** — min 0 — The length of one side of the square.
- **Diagonal Length (d)** — min 0 — The length of the diagonal of the square.

## Outputs

- **Diagonal** — The length of the diagonal: s x sqrt(2).
- **Side Length** — The length of one side: d / sqrt(2).
- **Area** — The area of the square.
- **Perimeter** — The perimeter of the square.

## Details

The diagonal of a square connects two opposite corners and is always longer than the sides. The relationship comes directly from the Pythagorean theorem.

**The Formula:**

d = s x sqrt(2)

Since a square's diagonal splits it into two right triangles with legs of length s, the diagonal is the hypotenuse: d = sqrt(s² + s²) = s x sqrt(2).

**Working Backwards:**

If you know the diagonal, the side length is s = d / sqrt(2), which equals d x sqrt(2) / 2.

**Area from the Diagonal:**

You can also find a square's area directly from its diagonal: A = d²/2. This works because A = s² and s = d/sqrt(2), so A = d²/2.

**Where This Comes Up:**

Square diagonals matter in construction (measuring if a frame is square), screen sizes (TV and monitor diagonals), woodworking (cutting diagonal braces), and even baseball (the distance between home plate and second base on a 90-foot diamond is 90 x sqrt(2) = about 127.3 feet).

## Frequently Asked Questions

**Q: What is the formula for the diagonal of a square?**

A: The diagonal of a square with side length s is d = s x sqrt(2), approximately s x 1.4142. This comes from the Pythagorean theorem applied to the right triangle formed by two sides and the diagonal.

**Q: How do I find the side length from the diagonal?**

A: Divide the diagonal by sqrt(2): s = d / sqrt(2). This is equivalent to s = d x sqrt(2) / 2. For example, a square with a diagonal of 10 has a side length of 10 / 1.4142 = approximately 7.071.

**Q: Can I find the area of a square using just the diagonal?**

A: Yes. The area of a square is A = d²/2, where d is the diagonal. For a square with a diagonal of 10, the area is 100/2 = 50. This formula comes from substituting s = d/sqrt(2) into A = s².

**Q: Why is the diagonal of a square equal to side times sqrt(2)?**

A: A square's diagonal creates two 45-45-90 right triangles. By the Pythagorean theorem, the hypotenuse (diagonal) equals sqrt(s² + s²) = sqrt(2s²) = s x sqrt(2). The ratio 1:1:sqrt(2) is a fundamental property of 45-45-90 triangles.

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Source: https://vastcalc.com/calculators/math/square-diagonal
Category: Math
Last updated: 2026-04-08