Area of Rectangle

The area of the rectangle is the region covered inside the boundaries of the rectangle. The area of a rectangle is calculated using its dimensions, length and breadth, similar to the square in which the side is both the length and breadth.

Properties of Rectangle

Some of the common properties of rectangles are

Formula

The area of a rectangle is the product of length (L) and breadth (B).

Area of rectangle Formula, A = L x B

Where,

• L is the length of the rectangle,
• B is the breadth of the rectangle

Note: If the unit of length and breadth is not exact then it should be transformed into one unit. For e.g. If the length is in cm and breadth in m then both dimensions should be adjusted either to m or cm.

Unit of Area of Rectangle

The area of a rectangle is measured in square units, and the standard unit for measuring the area of a rectangle is m². Other units widely used for measuring the area of the rectangle are cm², mm², and others.

Example: Find the area of a rectangle whose length is 20 inches and breadth is 50 inches.

Solution:

The formula for area of rectangle is given:

Area = L × B

⇒ Area = 20 × 50

⇒ Area = 1000 inches²

Thus, the required area is 1000 inches²

Area of Rectangle Formula Derivation

The area of a rectangle is the product of length and breadth. This can be derived by dividing the rectangle into two triangles. The triangles are equal, as the base and height of the two triangles will be equal. 

Area of Rectangle = 2 (Area of Triangle)

⇒ Area of Rectangle = 2 (1/2 × Base × Height)

⇒ Area of Rectangle = 2 (1/2 × AB × BC)

⇒ Area of Rectangle = AB × BC

⇒ Area of Rectangle = Length × Breadth.

Thus, the area of the rectangle formula is derived.

Area of Rectangle using Diagonal

The area of the rectangle can be found by two methods, which are:

Method 1 (Using Diagonal and One Side of Rectangle)

We can find the value of the missing side using the Pythagorean theorem and then find the area. Let us understand this using an example.

The diagonal of the rectangle is the line joining opposite vertices. The diagonal of the rectangle is calculated using Pythagoras's Theorem

(Diagonal)² = (Length)² + (Breadth)²

⇒ Length² = (Diagonals² - Breadth²)

⇒ Length = √(Diagonals² - Breadth²)

The formula for the area of a rectangle is calculated by

Area = Length × Breadth

⇒ Area = √(Diagonals² - Breadth²) × Breadth

⇒ Area = Breadth √(Diagonals² - Breadth²)

Method 2 (Using Area of Quadrilateral Formula)

If both the diagonals of the rectangle are given, then its area can be found with the help of the area of the quadrilateral formula.

Let a rectangle ABCD have diagonals as AC and BD, and let their lengths be d₁ and d2; then its area is given by,

Area of rectangle ABCD = 1/2 × d₁ × d₂

Example: Find the area of a rectangle whose length of the diagonals is 10 cm and 14 cm.

Solution:

The formula for area of rectangle is,

Area = 1/2 × d₁ × d₂

⇒ Area = 1/2 × 10 × 14

⇒ Area = 70 cm²

Thus, the area of required rectangle is 70 cm².

Area of Rectangle Using Perimeter

Follow the following steps to calculate the area of a rectangle using the perimeter and one side of the rectangle:

• Step 1: Note the perimeter and the given dimension.
• Step 2: Use the perimeter formula to find the other dimension.
• Step 3: Use the area of the rectangle formula and substitute the required value obtained in Step 2
• Step 4: Simplify the expression and add unit²  to get the final answer.

The example given below explains the above concept.

Example: Find the area of a rectangle when the perimeter is 28 cm and the breadth is 8 cm.

Solution:

Given,

Perimeter of Rectangle = 28 cm

length = 8 cm

breadth(b) = ?

Using Perimeter of rectangle formula,

Perimeter of rectangle = 2 (l + b)

⇒ 28 = 2 (8 + b)

⇒ 14 = 8 + b

⇒ b = 6 cm

Thus the breadth of rectangle is 6 cm

Area of Rectangle = l × b

⇒ Area of Rectangle = 8 × 6 = 48 cm²

Thus, the area of the Rectangle is 48 cm²

Also Check

• Area of Square
• Area of Triangle
• Area of Circle
• Area of Quadrilateral
• Area of Rhombus
• Area of Trapezium
• Area of Parallelogram

Solved Examples

Example 1: The length and width of a rectangle are 6 units and 3 units, respectively. Find the area of the rectangle.

Solution:

Given,  

• length = 6 units
• breadth = 3 units

Area of rectangle =  length × breath
⇒ Area of rectangle =  6 × 3
⇒ Area of rectangle = 18 square units

Thus, the area of given rectangle is 18 square units

Example 2: The height of a rectangular net is seen to be 20 cm. Its area is seen to be 260 cm². Find the width of the provided net.

Solution:

Given, 

Height = 20 cm
Area = 260 cm²

Area of Rectangle = width × height

⇒ width = Area / height

⇒ width = 260/20

⇒ width = 13 cm

Thus, the width of the rectangle is 13 cm

Example 3: The height and width of a rectangular desk are 40 m and 20 m, respectively. If a carpenter charges ₹ 2 per m² for his work, how much would it cost to make the whole desk?

Solution:

Given, 

Height of Desk = 40 m
Width of Desk = 20 m

Area of top of Desk = width of desk × height of desk

⇒ Area of top of Desk = 40 × 20

⇒ Area of top of Desk = 800 m²

At the cost of ₹ 2 per m², 

The cost for making top of the desk is 800 × 2 = ₹ 1600

Example 4: A wall whose length and width are 60 m and 40 m, respectively, needs to be painted. Find the quantity of the paint required if 1 liter of paint can paint 400 m² of the wall.

Solution:

Given, 

Length of wall = 60 m

Width of wall = 40 m

Thus, Area of wall = width × length

⇒ Area of wall = 60 × 40

⇒ Area of wall = 2400 m²

Paint required for 400 m² of wall = 1 litre   (given)

⇒ Paint required for 2400 m² of wall = 2400 / 400 × 1 = 6 litre.

Thus, the paint required to paint the wall is 6 litre.

Area of Rectangle Worksheet

Question 1: What is the area of a Rectangular Field of length 15 m and width 8 m?

Question 2: Find the Area of a Rectangle whose length is twice its breadth and perimeter is 72 cm.

Question 3: What is the cost of tiling a floor of length 10 m and width 11 m at the rate of 15 rupees per square meter?

Question 4: How many boxes of dimension 10 cm by 9 cm can be placed on a floor of 10 m by 9 m?

Using Diagonal and One side

Question 1: The diagonal of a rectangle is 13 cm, and one of its sides is 5 cm. Use the Pythagorean theorem to find the area of the rectangle.

Question 2: A rectangle has a diagonal of 17 cm and one side of 8 cm. Find the area of the rectangle.

Using Quadrilateral Formula

Question 1: Find the area of a rectangle that has two diagonals, each measuring 20 cm.

Question 2: Calculate the area of a rectangle whose diagonals measure 34 cm each.