Area of Rectangle - Definition, Formula, Derivation and Examples

A rectangle is a type of quadrilateral that has opposite sides that are equal in length and parallel to each other. It is classified as an equiangular quadrilateral because it has four right angles. Since rectangles have opposite sides that are parallel and equal in length, they can also be referred to as parallelograms. Rectangles exist in a two-dimensional space and can be represented on an XY plane by using their length and breadth as the x and y axes, respectively. A rectangle with all sides equal in length is known as a square. The area of a rectangle is the space that is contained within its boundaries and can be calculated by counting the number of unit squares that can fit inside it without overlap. Alternatively, the area of a rectangle can be found by multiplying its length and breadth together.

Length And Width Of A Rectangle

Length: The distance between two points is scaled using length. It is the size of a figure that indicates how long an object or figure is. Measurements, centimetres, elevation, and other direct units are used to express it.

Width: It is also used to scale the distance between two points. The shorter length of an object or figure is its width, which indicates how broad or wide the object or figure is. Additionally, the range is stated in direct units such as measurements, centimetres, elevation, etc.

Area Of Rectangle Using Length And Width

The area of a rectangle is conditioned by its length and width. The area of a rectangle is the product of its length and breadth (or width) and it is determined in square-units dimension.

Area = length ⨯ breadth

A = l ⨯ b

Whenever we write an area, it must be written in square meters.

If the rectangle is a square then the area becomes the square of the side length.

Derivation Of Area Of Rectangle

When we draw a diagonal of a rectangle it creates two right-angled triangles that are identical in shape and size. These triangles have the same base and height as the corresponding sides of the rectangle. The area of each triangle can be calculated using the formula (½ * base * height). Therefore, the total area of the rectangle can be found by adding up the areas of the two triangles, which is equal to the sum of their individual areas.

Area of rectangle 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 (length ⨯ breadth)

Area of rectangle = ½ (length ⨯ breadth) + ½ (length ⨯ breadth) = length ⨯ breadth

Area Of Rectangle Using Diagonal

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

(Diagonal)^2 = (length)^2 + (Breadth)^2

We can determine the area of a rectangle if the length of the diagonal and either length or breadth is known to us. From the formula of diagonal, we rewrite the length as:

(length)^2 = (Diagonal)^2 -(Breadth)^2 \\

length = \sqrt{(Diagonal)^2 -(Breadth)^2}

The area of a rectangle is equal to the product of length and breadth. Hence,

Area = length \times breadth \\

Area = (\sqrt{(Diagonal)^2 -(Breadth)^2}) \times breadth

Examples

1. The length and breadth of a book cover is 30 cm and 20cm respectively. What is the area of the book cover?

Solution: The book cover has the shape of a rectangle. The area of a rectangle is given by the following formula:

Area = length ⨯ breadth

Given, the length = 30 cm and breadth = 20 cm

Area = 30 cm ⨯ 20 cm

= 600 cm^2

2. The area of a rectangular farm is equal to 1000 square meters. It has a length of 40 meters. What is the width of the farm?

Solution: Area of rectangular farm = length ⨯ width

Given, area = 1000 sq. m and

Length = 40 m

1000 = 40 ⨯ width

Width = \frac{1000}{40} m\\

Width = 25 m

Hence, the farm is 25 m wide.