#### What is Square?

Square is the regular quadrilateral where all four sides are equal and angles are also same i.e. 90-degree each. This is a special type of rectangle where adjacent sides are equal.

The square is a special rhombus where all four sides are equal and opposite angles are equal. This is a parallelogram whose opposite sides are equal, this is a quadrilateral whose four sides are equal and a rectangle having opposite sides equal and aligned at 90-degree angle.

- The diagonals of a square will bisect each other at the right angle
- The diagonal of a square bisectits angles too.
- The opposite sides of a square are always parallel having equal in length.
- All the four angles in a square would be equal.
- All the four sides of a square will be equal.
- The diagonals of a square are also equal in length.

## Square Formulas

Square is a regular quadrilateral having all four sides and four angles are equal. Each side is right-angled and this is a special case of the rectangle where the adjacent sides are equal in length. To calculate the area, length of diagonals, and its perimeter, we need a list of basic square formula to make the computation easier.

\[\LARGE Area\:of\:a\:Square\: = a^{2}\]

\[\LARGE Perimeter\:of\:a\:Square\: = 4 \times a\]

\[\LARGE Diagonal\:of\:a\:Square\: = a\sqrt{2}\]

Where ‘a’ is the length of a side of the square.

### Area of a Square Formula

Geometrically, the square is a flat figure having all four sides equal and the angle is the right angle. It also states that opposite sites are parallel or this is a special case of the rectangle where adjacent sides are equal in length. The area is the total space occupied within the square.

This is a common quadrilateral in geometry whose area computation is part of our routine life too. It is also named as regular polygon due to four equal sides that are aligned at 90-degree angle each. So, properties of a regular polygon can also be applied to the square as well. Thus, the area of a square formula can be written as –

\[\LARGE Area\:of\:a\:Square\: = a^{2}\]

Where ‘a’ is the length of a side of the square.

The other popular term is the surface area of a square. People are usually confused between two terms area and the surface area. Do they both are the same? The surface area is the generally a three-dimensional figure like cube where more than one surface is available but in the case of a square, there is only one surface. In this case, the surface area of a square would always be equal to the area of a square. So, you need to use the same formula either it is the surface area of just area of a square.

### The perimeter of a Square Formula

The Perimeter is defined as the boundary length of a square. The perimeter of a square formula could be written as –

\[\LARGE Perimeter\:of\:a\:Square\: = 4 \times a\]

Where ‘a’ is the length of a side of the square.

**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:** a^{2} = 23^{2} = 529cm^{2}

**Perimeter of the square= **4a**= **4 × 23 = 92 cm

\[\LARGE Diagonal\:of\:a\:Square\: = a\sqrt{2} = 23\sqrt{2} = 32.52cm \]