Area of Rectangle - Definition, Formula, Derivation and Examples

A rectangle can be seen as a type of quadrilateral, a 2-D figure that has four sides and four vertices. When we talk about the other dimensions of figure-like angles, all the four angles of the rectangle are measured as 90 degrees with opposite sides being equal and parallel. In this article, we will learn how to calculate area of rectangle and its various applications in daily life.

What is Area of Rectangle

The area of rectangle in simple terms can be defined as the region occupied within the boundary of the rectangle. Or the other way round, the amount of surface enclosed by a rectangle is called the area of rectangle.

The area of a rectangle is dependent on its sides. The formula for the area of a rectangle can be stated as follows: It is equal to the product of the length and breadth of the rectangle. Whereas when we speak about the perimeter, it is equal to the sum of all its four sides. Hence, we can conclude that the region enclosed by the perimeter of the rectangle is the area of rectangle. However, lateral and total surface areas can be calculated only for three-dimensional figures and therefore it cannot be calculated for a rectangle since it does not fulfill this criteria.

Formula of Area of Rectangle

Area of rectangle = Length $\times$ Breadth

$
{A}={lb}
$

Few common examples include the flat surfaces of laptop monitors, blackboards, painting canvas, etc. In the case of a square, the area will become side ${ }^2$. The main difference between the two figures is that the length and breadth are equal for square

Derivation of Area of Rectangle Formula

A rectangle PQRS has a diagonal. Diagonal PR divides the rectangle PQRS into two congruent triangles The area of rectangle PQRS is the sum of the area of these two triangles.

Area of Rectangle PQRS = Area of Triangle PQR + Area of Triangle PRS
Since both the triangles (PQR and PRS) are congruent, we can express it as follows.
Area of Rectangle PQRS $=2 \times$ Area of Triangle PQR
Area of Rectangle PQRS $=2 \times(1 / 2 \times$ Base $\times$ Height $)$
Area of Rectangle PQRS $=2 \times 1 / 2 \times P Q \times Q R$
Area of Rectangle $P Q R S=P Q \times Q R$
Hence, Area of Rectangle $=$ Length $\times$ Width

How to find Area of Rectangle

We follow the below-mentioned steps to find the area of the rectangle:

Step 1: We write down the dimensions of length and width from the data.

Step 2: Next, we multiply length and width.

Step 3: Finally we write the answer in square units.

Surface Area of Rectangle

We can use the following steps to find the surface area of the rectangle:

Step 1: First we multiply the length ($l$) by the width ($b$) of rectangle.

Step 2: Hence we come across the expression $l \times b$, where $l$ is the length and $b$ is the width of rectangle.

We can say that surface area of rectangle is same as that of area of rectangle.

Area of Rectangle Using Diagonal

The area of a rectangle can be calculated if we know the diagonal and one side. We initially define diagonal of a rectangle as the straight line inside the rectangle connecting its opposite vertices. There are two diagonals in a rectangle and both of them are equal.

^{Example: Find area of rectangle whose length is 2 cm and whose diagonal is 3 cm.
Solution:
The width of the rectangle is missing and it can be calculated using the Pythagoras theorem because the diagonals of a rectangle form 2 right-angled triangles.}

^{In this case, the width can be calculated using the formula, width $=\left[\sqrt{(\text { Diagonal })^2-(\text { length })}^2\right]$
After substituting the given values, we get}

^{$
\begin{aligned}
& \text { width }=\sqrt{(3)^2-(2)^2} \\
& \text { width }=\sqrt{9-4 } \\
& \text { width }=\sqrt{5 } =2.23 \mathrm{~cm}
\end{aligned}
$}

^{Now, we know that the length $=2 \mathrm{~cm}$, width $=2.23 \mathrm{~cm}$. So, the area of the rectangle $=l \times b$. In this case,}

^{$
A=l \times b
$}

^{Area $=2 \times 2.23=6.44 \mathrm{~cm}^2$}

Area of Rectangle Using Perimeter

With the help of perimeter, we can find the unknown side and then find area using the same formula, Area of rectangle formula $=$ Length $\times$ Width. The illustration is explained below:

Example: Find the area of rectangle if the perimeter is 20 units and the length is 6 units.
Solution: Perimeter of the rectangle $=20$ units, length $=6$ units. We can find the width using the formula,

Perimeter of rectangle $=2(1+w)$
$20=2(6+w)$
$10=6+w$
$w=4$ units
width $=4$ units, length $=6$ units,
Area of rectangle $=1 \times w$
Area $=4 \times 6=24$ square units

Solved Examples Based on Area of Rectangle

Example 1: Find the area of rectangle whose length is 12 cm and the width is 2 cm.
Solution:
Given,
Length $=12 \mathrm{~cm}$
Width $=2 \mathrm{~cm}$
Area of rectangle formula $=$ Length $\times$ Width

$
12 \times 2=24
$

So the area of rectangle $=24 \mathrm{~cm}^2$

Example 2: What is the area of rectangular blackboard whose length and breadth are 200 cm and 100 cm, respectively.

Solution:
Length of the blackboard $=200 \mathrm{~cm}=2 \mathrm{~m}$
Breadth of the blackboard $=100 \mathrm{~cm}=1 \mathrm{~m}$
Area of the blackboard $=$ area of rectangle $=$ length x breadth $=2 \mathrm{~m} \times 1 \mathrm{~m}=2$ square-metres

Example 3: The length of a rectangular screen is 10 cm. Its area is $80 \mathrm{sq}. \mathrm{cm}$. Find its width.
Solution:
Area of the screen $=80 \mathrm{sq} . \mathrm{cm}$.
Length of the screen $=10 \mathrm{~cm}$
Area of rectangle formula $=$ length x width
So, width = area/length
Thus, width of the screen $=80 / 10=8 \mathrm{~cm}$

Example 4: The length and breadth of a rectangular wall are 70 m and 30 m, respectively. Find the cost of painting the wall if the rate of painting is Rs 3 per sq. m.

Solution:

Length of the wall $=70 \mathrm{~m}$
The breadth of the wall $=30 \mathrm{~m}$

Area of rectangle $=$ length $x$ breadth $=70 \mathrm{~m} \times 30 \mathrm{~m}=2100$ sq. m
For $1 \mathrm{sq}. \mathrm{m}$ of painting costs Rs 3
Thus, for 2400 sq. m, the cost of painting the wall will be $=3 \times 2100=$ Rs 6300

Example 5: A floor whose length and width is 60 m and 20 m respectively needs to be covered by rectangular tiles. The dimension of each tile is $1 \mathrm{~m} \times 2 \mathrm{~m}$. Find the total number of tiles that would be required to fully cover the floor.

Solution:
Length of the floor $=60 \mathrm{~m}$
The breadth of the floor $=20 \mathrm{~m}$
Area of the floor=area of rectangle $=$ length $x$ breadth $=60 \mathrm{~m} \times 20 \mathrm{~m}=1200 \mathrm{sq} . \mathrm{m}$
Length of one tile $=2 \mathrm{~m}$
The breadth of one tile $=1 \mathrm{~m}$
Area of one tile $=$ length $x$ breadth $=2 \mathrm{~m} \times 1 \mathrm{~m}=2 \mathrm{sq} . \mathrm{m}$
No. of tiles required $=$ area of floor/area of a tile $=1200 / 2=600$ tiles

Summary

We define area as the number of unit squares that can fit into any shape. The term “unit” refers to one, and a unit square is a square having one 1 unit of side. As a result, the area of a rectangle is equal to the number of unit squares within the rectangle’s edge. Area of rectangle formula helps us to do calculations easily. In this article we have learnt about area of rectangle, formula for area of rectangle, surface area of rectangle. We have also covered the concept of perimeter and area of rectangle. They are all around us in many forms. The flat surfaces of laptop monitors, blackboards, painting canvas, and other rectangular shapes are examples.