# Diagonal of a Square Calculator

Created by Hanna Pamuła, PhD
Reviewed by Bogna Szyk and Adena Benn
Last updated: Jan 31, 2023

Find out the length of the square's diagonal with our diagonal of a square calculator. Check out below how to find the diagonal of a square formula, or simply give our tool a try - you won't be disappointed.

## Diagonal of square

A diagonal is a line segment that joins two non-neighboring vertices. Each quadrilateral has two diagonals, a square too. Its diagonals are:

• Equal in length;
• Perpendicular bisectors of each other; and
• Bisectors of square angles.

Each diagonal divides the square into two congruent isosceles right triangles — we made the 45 45 90 triangle calculator about them. Such a triangle has half of the area of a square, its legs are square sides, and the hypotenuse equals the length of the diagonal of a square.

## What is the formula for the diagonal of a square?

To calculate the length of the diagonal of a square, multiply the length of the side by the square root of 2: diagonal = √2 × side. Ingrain this relationship into your brain and heart so that never again will you wonder how to find the diagonal of a square.

Let's analyze an example. If the square side is equal to 5 in, then the diagonal is $5\sqrt{2}\ \text{in} \approx 7.071\ \text{in}$. Type that value into the diagonal of a square calculator to check it yourself!

Where does this equation come from? You can derive this diagonal of square formula e.g., from calculating the Pythagorean theorem.

1. As we noticed before, the diagonal of a square divides the square into two congruent right triangles.
2. Legs are square sides, and the length of the diagonal of the square is the hypotenuse. Given the side length of the square:
$\begin{split} &a^2+a^2=\text{diagonal}^2\\ &\text{diagonal} = \sqrt{a^2+a^2}=\sqrt{2\cdot a^2}\\ &\text{diagonal}=a\cdot\sqrt{2} \end{split}$

This is the same as calculating the hypotenuse in a right triangle.

If you don't have the side of a square given, use other formulas:

$d = \sqrt{2\cdot\text{area}}$

If the area is given, and:

$d = \frac{\text{perimeter}}{4}\cdot\sqrt{2}$

If we know the square's perimeter.

🙋 We can also teach you how to find the diagonal of a rectangle with our diagonal of a rectangle calculator!

## How to use this diagonal of a square calculator?

With our calculator, it's a piece of cake!

1. Type the given value into a proper box. Let's say we know that the square side is equal to 8 in.
2. Diagonal of a square calculator displays the result! In our case, the diagonal is $15.556\ \text{in}$.
3. If you want to check the result in a different unit, click on the unit name and select the one that meets your needs. For instance, $15.556\ \text{in}$ is $39.51\ \text{cm}$ and around $1\ \text{ft}\ 3\ \text{in}$.
4. The diagonal of a square calculator works the other way round as well - knowing the diagonal, you can calculate the square side.

Now you're the expert, and you know exactly how to find the diagonal of a square given square sides. However, if you don't have it provided, use this general square calculator, where you can type the area or perimeter, and the tool will also find the diagonal.

## FAQ

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

To compute the length of a side of a square:

1. Write down the length of the square's diagonal d.
2. Divide d by √2, or, if an approximation is enough, by 1.4142 or simply by 1.41.
3. That's it! If you struggle with the calculations, use an online square calculator.

### What is the diagonal of a square of side 1?

The answer is √2, that is, about 1.41. This is because the diagonal and side are connected by the formula diagonal = √2 × side. Plugging in the value side = 1, we arrive at diagonal = √2. It now suffices to recall that √2 is equal to 1.4142....

### What is the side of a square of diagonal 1?

The side has length 0.707, or, more precisely, √2/2. To arrive at this answer, you need to use the formula side = diagonal / √2. As we know that diagonal = 1, we easily get side = 1 / √2. To simplify, multiply both the numerator and denominator by √2. This gives side = √2 / 2 ≈ 0.707.

Hanna Pamuła, PhD
Side a
in
Diagonal d
in
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