Diagonal of Square Formulas | List of Diagonal of Square Formulas You Should Know - BYJUS

# Diagonal of Square Formulas

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

### Square

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.

### Properties of Diagonals

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

### How is the Formula for the Diagonal of a Square derived?

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)

$$d=a\sqrt2$$

### Formula for the Diagonals of a Square

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.

### Solved Examples

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.