# Square Shape

By the end of this lesson, you will be able to

- define square.
- calculate the perimeter of a square.
- determine the diagonal of a square.
- evaluate the area of a square.
- relate square to rectangle, rhombus and parallelogram.
- represent examples of square.

## Square

A square is a quadrilateral in which all four sides and angles are equal. As the sum of four interior angles is 360° and each angle is equal to each other, each interior angle of a square is equal to 90°.

A square is in figure.

Again, since each angle of a square is a right, it is sometimes called a right quadrilateral. It is regular quadrilateral as well as it is a regular polygon.

## Apps for Finding Area, Perimeter and Diagonal

Perimeter:

Area:

Diagonal:

Drag square or yellow area and see the change.

## Square Formula

All the formulae that are used in generally to solve problems invoving square are given below.

## Perimeter of Square

The sum of the length of four sides of a square is called the perimeter of a square. So, the perimeter of a square means the total length of outline of a square. As the length of the side of a square is equal to each other, the perimeter is the four times the length of the side of square.

### Perimeter of Square Formula

Let AB = BC = CD = AD = a be the length of the side and P be the perimeter of square ABCD.

So, the perimeter of the square,

P = (AB + BC + CD + AD) unit

বা, P = (a + a + a + a) unit

∴ P = 4a unit

So, perimeter of square = 4×a unit

If the length of the side and perimeter are a and P of a square respectively,

P = 4a unit

## Diagonal of Square

The line segment connecting two opposite vertices of a square is called the diagonal of a square. And length of this line segment is called the length of the diagonal. There are two diagonals of a square. The two diagonals are of equal length. Each diagonal divides the square into two congruent triangle. Again, each triangle is a right triangle as the each angle of a square is 90°.

A square is showing its equal diagonals.

The area of each triangle is half the area of the square. Moreover, two diagonals bisect at right angle to one another. The is wonderful characteristics of a square. A square is a regular polygon as well as a regular quadrilateral.

### Diagonal of Square Formula

Let a and d be the length of the side and diagonal of a square. The two sides a and a; and the diagonal d form a right triangle whose hypotenuse is d and the legs (base and height) are a and a. Applying Pythagorean theorem, it can be as,

d^{2} = a^{2} + a^{2}

or, d^{2} = 2a^{2}

or, d = √2a^{2}

or, d = a√2

If the side and the diagonal are a and d respectively,

d = a√2

### Square Example

## Area of Square

The square of the length of the side of a square is called the area of a square. The area of a square means how much interior surface it contains bu its four sides. Area is related to 2-dimensional geometry.

### Area of Square Formula

It is known to us that the area of a rectangle is the product of length and width of a rectangle. A square is always a square. So, the area formula for rectangle can be applied to square. Therefore, the area of a square is equal to the product of a couple of adjacent sides.

So, area of square = (length × length) square unit

Let the length of the side and the area of a square be a and A respectively.

So, A = (a × a) square unit

∴ A = a^{2} square unit

If the length of the side is a and the area is A,

A = a^{2} square unit