Area of Square Formula and Solved Examples

A quadrilateral with all sides equal and parallel opposite sides is known as a square. It is the only regular polygon whose diagonals are all the same length and whose internal, central, and external angles are all identical and equal to 90 degrees. Squares can also be referred to as rectangles with equal adjacent sides. Squares are flat, two-dimensional shapes. We may easily represent a square in an XY plane by using its sides as the arms of the x and y axes. The quantity of unit squares contained within a square's perimeter is its area. Alternatively, the area of the square is the area contained within the boundary of this shape.

Area Of The Square Formula Using Its Side Length

The area of a square is conditioned on the length of its side. The area of a square is equal to the square of its side length and it is determined in square-units dimension.

Area = side^2

We must always write the area in square units.

Derivation Of Area Of Square Formula

The diagonals of a square divide it into two congruent right-angled triangles with base and height equal to the sides of the square. The area of the square will be equal to the sum of the areas of individual areas of triangles.

Area of square ABCD = Area of triangle ABC + Area of triangle ADC

The area of both triangles is the same since they are congruent triangles. The area of triangles is half of the product of the base and height.

Area of triangle ABC = Area of triangle ADC = ½ (side ⨯ side)

Area of square = ½ (side ⨯ side) + ½ (side ⨯ side)

= side^2

Area Of Square Formula Using Diagonal

The straight line connecting the opposite vertices of a square is called the diagonal of a square. There are two diagonals in a square and the measure of the length of both diagonals is equal. The length of diagonals can be determined using the length of the side of a square by applying the Pythagoras theorem. Hence, the formula for diagonal is given by:

(Diagonal)^2 = (side)^2 + (side)^2 = 2(side)^2

We can determine the area of a square if the length of the diagonal is known to us. From the formula of diagonal, we rewrite the side as:

(side)^2 = \frac{(Diagonal)^2}{2} \\

side = \frac{Diagonal}{\sqrt{2}}

The area of a square is equal to the square of the side. Hence,

Area = side^2 \\

Area = (\frac{Diagonal}{\sqrt{2}})^2\\

Area =\frac{(Diagonal)^2}{2}

Examples

1. The length of the sides of a square tile is 30 cm. What is the area of the tile?

Solution: The area of a square is given by the following formula:

Area = side^2

Given, the side = 30 cm

Area = 30 cm ⨯ 30 cm

= 900 cm^2

2. The area of a square farm is equal to 1600 square meters. What is the length of the sides of the farm?

Solution: Area of square farm = side^2

Given, area = 1600 sq. m

1600 = side^2

side = \sqrt{1600}\\

= 40

Hence, the length of the sides of the farm is 40 m.