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Area of Square Formula and Solved Examples

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Area of Square Formula and Solved Examples

Edited By Team Careers360 | Updated on Jan 29, 2024 10:32 AM IST

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

1706367730204

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 1706367730471

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

1706367730004

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}}

1706367730726

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}

1706367730121

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 1706367730570

Given, the side = 30 cm

Area = 30 cm ⨯ 30 cm

= 900 cm^2 1706367730793

  1. 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 1706367730324

Given, area = 1600 sq. m

1600 = side^2 1706367730407

side = \sqrt{1600}\\

= 40

1706367730843

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

Frequently Asked Question (FAQs)

1. What is a square?

A square is a two-dimensional shape. It is a quadrilateral with all equal sides and parallel opposite sides. 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.

2. What is the area of a square and how to calculate it?

The area of the square is the area contained within the boundary of this shape. Alternatively, the quantity of unit squares contained within a square's perimeter is its area. The area of a square is equal to the square of its side.

Area = side^2 

3. What is the perimeter of a square?

The whole distance that a square's boundaries or its sides cover are known as its perimeter.  In essence, the perimeter reveals the length of the figure. Given that a square has four sides, the perimeter of the square will be equal to the sum of its four sides. The unit of the perimeter of a square will be in meters, centimetres, inches, etc. since the perimeter is a linear measurement. 

The perimeter of a square = 4✖(side)

4. If the side length of a square is made thrice the original side length then how will the area of the square change?

The area of a square is directly proportional to the side length’s square. On making the side three times the original one, the area will become nine times the original area which is equal to the square of three.

5. Can we calculate the side of a square if its area is known to us? If yes, how?

Yes, we can determine the side of a square if the area is known to us because the area of a square is dependent on the side square. We can find the side by using the following formula:

Side = \sqrt{area}

Area - Definition, Area of Shapes Formula

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