Square is a regularÂ quadrilateral. All the four sides and angles of a square are equal. The four angles are 90 degrees each, that is, right angles. A square may also be considered as a special case of rectangle wherein the two adjacent sides are of equal length.

In this section, we will learn about the square formulas â€“ a list of the formula related to squares which will help you compute its area, perimeter, and length of its diagonals. They are enlisted below:

**\(Area\; of\; a\; Square\; = a^{2}\)**

**Perimeter of a Square =4a**

**\(Diagonal\; of\; a\; Square\; = a\sqrt{2}\)**

Where â€˜aâ€™ is the length of a side of the square.

**Properties of a Square**

- The lengths of all its four sides are equal.
- The measurements of all its four angles are equal.
- The two diagonals bisect each other at right angles, that is, 90Â°.
- The opposite sides of a square are both parallel and equal in length.
- The lengths of diagonals of a square are equal.

**Derivations:**

Consider a square with the lengths of its side and diagonal are *a* and *d *units respectively.

**Formula for area**Â **of a square:** AreaÂ of a square can be defined as the region which is enclosed within its boundary. As we mentioned, a square is nothing a rectangle with its two adjacent sides being equal in length. Hence, we express area as:

The area of a rectangle = Length Ã— Breadth

Here,

**\(Area\; of\; square\; = a \times a = a^{2}\)**

**The formula for the perimeter**Â **of a square:** Perimeter of the square is the length of its boundary. The sum of the length of all sides of a square represents its boundary. Hence, the formula can be given by:

Perimeter = length of 4 sides

Perimeter = a+ a + a + a

**Perimeter of square = 4a**

**The formula for diagonal**Â **of a square:** A diagonal is a line which joins two opposite sides in a polygon. For calculating the length diagonal of a square, we make use of the Pythagoras Theorem.

In the above figure, the diagonalâ€™ divides the square into two right angled triangles. It can be noted here that since the adjacent sides of a square are equal in length, the right angled triangle is also isosceles with each of its sides being of length â€˜aâ€™.

Hence, we can conveniently apply the Pythagorean theorem on these triangles with base and perpendicular being â€˜aâ€™ units and hypotenuse beingâ€™ units. So we have:

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

**Or **

**\(d=\sqrt{2a^{2}}=a\sqrt{2}\;units\)**

**Solved examples:**

**Question 1:** A square has one of its sides measuring 23 cm. Calculate its area, perimeter, and length of its diagonal.

**Solution:** Given,

Side of the square = 23 cm

**Area of the square:**

Area of the square = *\(a^{2}=23^{2}=529 cm^{2}\)*

**Perimeter of the square: **

**Perimeter of the square= **4a**= **4 Ã— 23 = 92 cm

**Diagonal of a square: **

Diagonal of a square = \(a\sqrt{2}=23\sqrt{2\;cm}=32.52cm\)

**Question 2: Â **A rectangular floor is 50 m long and 20 m wide. Square tiles, each of 5 m side length, are to be used to cover the floor. Find the total number of tiles which will be required to cover the floor.

**Solution**: Given,

Length of the floor = 50 m

Breadth = 20 m

Area of the rectangular floor = length x breadth = 50 m x 20 m = 1000 sq. m

Side of one tile = 5 m

Area of one such tile = side x side = 5 m x 5 m = 25 sq. m

\(No.\; of\; tiles\; needed = \frac{Area\; of\; floor}{Area\; of\; one\; tile}=\frac{1000}{25}=40\; tiles\)

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