## What Is the Diagonal of a Square?

**Diagonals of a square are the line segments joining opposite vertices.**

A flat, closed shape with four equal sides, four vertices (like craft paper, a photo frame), and four right angles is known as a square. Take a look at the square craft paper shown below. Let’s connect two opposite corners. The two lines are called diagonals of a square.

### Diagonal of a Square Definition

**The diagonal of a square is a line that connects one corner to the opposite corner through the center.** In other words, we can say that the diagonal is the slant line that connects the square’s opposite corners.

A square has two diagonals that are equal in length. They bisect each other at right angles.

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## Properties of the Diagonal of a Square

- The diagonal of a square is a line segment that connects any two non-adjacent vertices (corners).

Here, AC and BD are diagonals.

- The diagonals of a square are equal in length.

$AC = BD$

- The square is divided into two congruent right-angle isosceles triangles by its diagonal.

- Two diagonals of a square bisect each other at right angles.

- When the diagonals meet the vertices of a square, they form a 45-degree angle. In other words, they bisect each pair of opposite angles.

- Pythagoras’ theorem can also be used to determine the length of the diagonals of a square.

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## Diagonal of a Square Formula

The formula for the diagonal of a square is given by the product of side length and the square root of 2.

Mathematically, the diagonal of a square formula is given by

$d = a\times\sqrt{2} = \sqrt{2} a$

where,

$d =$ length of diagonal of a square

$a =$ side length of the square.

## Derivation of Diagonal of Square Formula

Let’s derive the diagonal of a square formula.

Let us assume that the side length of a square is “a” and the length of the diagonal is “d.”

In the above figure, a square is divided into two right-angle triangles. Let’s consider triangle ABC.

In $\Delta ABC,\; \angleB = 90^\circ$.

Note that the diagonal AC acts as a hypotenuse of the right triangle ABC.

So, now in $\Delta ABC$ apply Pythagoras’ theorem,

$AC^2 = AB^2 + BC^2$

$d^2 = a^2 + a^2$

$d^2 = 2a^2$

$d = \sqrt{2a^2}$

$d = a\sqrt{2}$

Hence, diagonal “d” of square is given by $d = a\sqrt{2}$

where “d” is the diagonal length and “a” is the side-length of the square.

## How to Find the Diagonal of a Square

How to calculate the length of diagonal of a square? What is the diagonal measurement of a square? Finding the diagonal of a square is simple once you know the formula! Use the diagonal of a square formula. Multiply the length of the side by $\sqrt{2}$.

**Length of the diagonal of a square **$=$** Length of the side of the square **$\times\sqrt{2}$

To get an estimate, we can use $\sqrt{2}\simeq 1.41$.

## Methods to Calculate Diagonal of Square

The diagonal of a square can be calculated in three different ways.

**Method 1:** When the Side-Length of Square Is Given

If the length of one side of the square is known to us, then we can simply calculate the diagonal of a square using the formula $d = a\sqrt{2}$.

Where, d = length of diagonal of a square and a $=$ side length of the square.

**Method 2: **When the Perimeter of Square Is Given

If the perimeter of a square is given or known to us, then follow the below steps

- First find the side length (a) of the square by using the perimeter of the square formula.

Perimeter of a square$ = 4\times a$.

Side of the square $= \frac{Perimeter}{4}$

- Calculate the diagonal of the square using the formula: $d = a\sqrt{2}$

**Method 3: **When the Area of a Square Is Given

If the area of a square is given or known to us, then follow the below steps:

- First find the side length (a) of the square by using the area of square formula.

Area of a square $= a^2$

Side of the square $= \sqrt{Area} = a$

- Calculate the diagonal of the square using the formula $d = a\sqrt{2}$.

## Facts about Diagonals of a Square

- The diagonal of a square is longer than its side.

- A square has four lines of symmetry, one vertical, one horizontal, and two diagonal lines.

## Conclusion

In this article, we learned about the diagonal of a square formula, its properties, formula, derivation, different methods to calculate the length of a diagonal if the side length is given, and related facts, examples, and practice problems based on the same.

## Solved Examples on Diagonals of a Square

**Find the diagonal length of a square with 18 inches of side.**

**Solution:**

The side length of the square “a” $= 18$ inches.

According to the diagonal of square formula,

The diagonal length “d” of the square $= a\sqrt{2} = 18\times\sqrt{2} = 18\times1.414 = 25.42$

Hence, the diagonal length of the square is 25.42 inches.

**The length of the diagonal of a square is**$15\sqrt{2}$**inches. Find the side length of its side.**

**Solution:**

The diagonal length of the square $= d = 15\sqrt{2}$** **inches.

Let us consider the side length of the square to be “a.”

The diagonal length “d” of the square $= a\sqrt{2}$

$15\times\sqrt{2} = a\sqrt{2}$

Comparing both sides, we get

$a = 15$

Hence, the side length of the square is 15 inches.

**Find the diagonal length of a square field with a perimeter of 124 feet.**

**Solution:**

The perimeter of the square $= 124$ feet

Let us consider the side length of the carrom board to be “a.”

As we know, the perimeter of the square $= 4a$

$124 = 4a$

$a = \frac{124}{4} = 31$

Thus, $a = 31$ feet

Diagonal length $= d = a\sqrt{2} = 31\times\sqrt{2} = 43.83$ feet.

Hence, the diagonal length of the square field is 43.83 feet.

**Find the length of the diagonal of a craft paper if its area is 196 square units.**

**Solution:**

The Area of the craft paper $= 196$ square units

As we know craft paper is in the shape of a square.

The area of a square $= a^2 = 196$

Side length $= a = \sqrt{196} = 14$ units.

The diagonal length of the square $= d = a\sqrt{2} = 14\times\sqrt{2} = 19.8$ units.

Hence, the length of the diagonal of the craft paper is 19.8 units.

## Practice Problems on Diagonals of a Square

## Diagonal of a Square - Definition, Formula, Properties, Examples

### Which of the following objects do not come in a square shape?

Chess boards, floor tiles, and pizza boxes come in square shape but a yoga mat does not have the shape of a square.

### What is the total number of diagonals in a square?

Total number of diagonals in a square is 2.

### When the diagonals of a square bisect each other at a ____ angle.

The diagonals of a square bisect each other at a right angle, $90^\circ$.

### The length of the diagonal of a square field is $12\sqrt{2}$ feet. What is the length of its sides?

The length of the diagonal of a square field $= a\sqrt{2} = 12\sqrt{2} \Rightarrow a = 12$.

Hence, the length of its side $= 12$ feet.

### What is the length of each diagonal in a square with a side length of 5 inches?

The length of the diagonal of a square field $= a\sqrt{2} = 5\sqrt{2}$

Hence, the length of each diagonal in a square with a side length of 5 inches $= 5\sqrt{2}$ inches.

## Frequently Asked Questions on Diagonals of a Square

**Is the side of a square and its diagonal the same length?**

No, the side of a square and its diagonal aren’t of the same length. The diagonal of a square is greater in length than its side.

**Which quadrilaterals have diagonals that bisect each other?**

The quadrilaterals having diagonals that bisect each other: a rectangle, square, parallelogram, and rhombus.

**How to calculate the diagonal of a square when its side is given?**

If the length of side of the square is given, then we can calculate the diagonal of a square using the formula $d = side\times\sqrt{2}$.

**How to find the length of the diagonal of a square when its perimeter is given?**

If the perimeter of the square is given then first calculate the side length using the formula

“a” $= \frac{perimeter}{4}$ , then substitute the value of “a” in the diagonal length of a square formula $d = a\sqrt{2}$

$\Rightarrow d = \bigg(\frac{perimeter}{4}\bigg)\sqrt{2}$

**How to find the length of the diagonal of a square when its area is given?**

If the area of the square is given then first calculate the side length using the formula “a” $= \sqrt{Area}$ then substitute the value of “a” in the diagonal length of a square formula $d = a\sqrt{2}$

$\Rightarrow d = \sqrt{Area}\times\sqrt{2}$.