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A line segment that connects any two non-adjacent vertices forms the diagonals of a square. Two diagonals of equal length that intersect each other at right angles can be drawn inside a square. When the length of the side is known, the diagonal of the square can be calculated using a specific formula....Read MoreRead Less

In geometry, a square is a regular quadrilateral that has four equal sides and four angles each of which are 90\(^{\circ}\) in measure. As a result, the sum total of the internal angles in a square is 360\(^{\circ}\). A square has two diagonals, and as seen earlier, are created by connecting the opposing vertices of a square.

In order to list the qualities of diagonals, consider the following square shown in the image.

- The diagonals of a square are of equal length
- Diagonals of a square bisect each other perpendicularly
- They divide the square into two identical, right-angled triangles using isosceles triangles

- The mid point of both diagonals is where they intersect, which means that both diagonals bisect each other into two halves

Let’s take the triangle ADC. Since, we are aware that every angle in a square is 90° degrees, we can apply the Pythagoras theorem to determine the hypotenuse, which in this instance is ‘\(d\)’.

\(d^2=a^2\ +\ a^2\)

\(d=\sqrt{a^2+a^2}\) (Bringing the square from L.H.S to R.H.S)

\(d=\sqrt{{2a}^2}\) (Simplified)

\(d=a\sqrt2\)

Diagonal of a square formula is, \(d=a\sqrt2\)

Where ‘\(a\)’ is a side of the square and ‘\(d\)’ is the diagonal.

The Pythagorean theorem is used to calculate the formula for the diagonal of a square.

As seen earlier on in this article, the diagonal creates two isosceles right-angled triangles in a square. Both diagonals cut each other at right angles and are congruent.

**Example 1: If a square has an area of 25 square units, determine the length of its diagonal?**

**Solution:**

Given, area of the square = 25 square units

\(a^2=25\) [Use the formula for the area of a square]

Therefore, \(a=\sqrt{25}=5\) units

Since, \(d=a\sqrt2\) [Use the formula of diagonals of a square]

\(=a\sqrt2=5\sqrt2=7.071\) units

Hence, the length of the diagonal of a square is 7.071 units.

**Example 2: If a square has an area of 64 square units, determine the length of the diagonal.**

**Solution:**

Given: Area of the square = 64 square units

\(a^2=64\) [Use the formula for the area of a square]

Therefore, \(a=\sqrt{64}=8\) units

Since, \(d=a\sqrt2\) [Use the formula of diagonals of the square]

\(=a\sqrt2=8\sqrt2=11.312\) units

Hence, the length of the diagonal of the square is 11.312 units.

**Example 3: Rita has a square sheet of paper with a side length of 4 inches. She wants to decorate the diagonals of this sheet with colorful sticker paper. What is the total length of the sticker paper she will need for both the diagonals?**

**Solution:**

Given: Side of the square = 4 inches

Since, \(d=a\sqrt2\) [Use the formula of diagonals of the square]

\(=a\sqrt2=4\sqrt2=5.656\) inches

Hence, the length of the sticker paper needed is twice the length of one diagonal, \(5.656\times 2 = 11.31\) or approximately 11.5 inches.

Frequently Asked Questions

Every square has two diagonals that are of equal length. These diagonals also bisect each other at right angles.

The diagonals are the longest lines in a square.

In every right triangle we observe that the hypotenuse is the longest side of a triangle, and the diagonal is the hypotenuse of a triangle formed by two sides and the diagonal. Hence the side length cannot be longer than the diagonals of a square.

The diagonal is root two times the side length of a square.