Area of Rectangle

The area of a rectangle is the region occupied within the boundary of the rectangle. In other words, the amount of surface enclosed by a rectangle is called the area of the rectangle. This can be calculated using the area of rectangle formula and using various methods depending on the given dimensions. Let us learn how to find the area of a rectangle in this lesson.

The area of a rectangle is the number of unit squares that can fit into a rectangle. Area is defined as the amount of space covered by a flat surface of a particular shape. The area of any shape is the number of unit squares that can fit into it and it is measured in terms of the 'number of' square units (square centimeters, square inches, square feet, etc.). Some examples of rectangular shapes are the flat surfaces of laptop monitors, blackboards, painting canvas, etc. The area of a rectangle can be calculated by using the area of rectangle formula to find the space occupied by these objects.

Area of a Rectangle Definition

The region occupied by a rectangle within its boundary is defined as the area of the rectangle.

The formula of the area of a rectangle is used to find the area occupied by the rectangle within its boundary. The area of a rectangle is obtained by multiplying its length and width (breadth). Thus, the formula for the area of a rectangle whose length and width are 'l' and 'w' respectively is expressed as follows. Area of rectangle = l × w. In other words, Area of a Rectangle = (Length × Width)

The area of a rectangle is the product of its length and width (breadth). The area of a rectangle can be calculated using the steps given below:

• Step 1: Note the dimensions of length and width (breadth) from the given data.
• Step 2: Find the product of the length and width values.
• Step 3: Give the answer in square units.

Let us take an example to understand the calculation of the area of a rectangle.

Example: Find the area of the rectangle whose length is 15 units and width is 4 units.

Solution: Given, length = 15 units and width = 4 units.

The formula to find the area of a rectangle is A = l × w. Substitute 15 for 'l' and 4 for 'w' in this formula. Therefore, area of the rectangle = 15 × 4 = 60

Therefore, the area of the rectangle = 60 square units.

Unit of Area of Rectangle

The unit of the area of a rectangle is expressed in square units. We know that length is always measured and expressed in units like cm, inches, etc. Now, let us understand the reason for which the area of a rectangle is expressed in square units using the following example.

Example: The length of a rectangle is 4 inches and its width is 3 inches. Observe the figure given below in which we can divide the figure into 12 small squares, each of which is a square, that is, 1 inch on each side, that is, 1 square inch. Since 12 equal-sized squares can fit in this rectangle, they also show the space occupied by the whole rectangle. So, a rectangle whose sides are 4 inches and 3 inches has an area of 12 square inches or 12 in². In other words, 12 unit squares can fit in the given rectangle covering all its space and this is termed as the area of a rectangle.

In this case, since the length and width of this rectangle is given in 'inches', the area is measured and written in square inches (in²). Similarly, if the length and width of any rectangle is given in cm, then the area will be expressed in square centimeters (cm²).

The area of a rectangle can be calculated if the diagonal and one side is given. The diagonal of a rectangle is the straight line inside the rectangle connecting its opposite vertices. There are two diagonals in a rectangle and both are of equal length. There can be two ways in which we can find the area of a rectangle using the diagonal.

Method 1:

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

Example: Find the area of a rectangle whose length is 4 cm and whose diagonal is 5 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 = ⎷[(Diagonal)² - (length)²]

After substituting the given values, we get, width = ⎷[(Diagonal)² - (length)²]

width = ⎷[(5)² - (4)²]

width = ⎷(25 - 16)

width = ⎷9 = 3 cm

Now, we know that the length = 4 cm, width = 3 cm. So, the area of the rectangle = l × w. In this case,

A = l × w

Area = 4 × 3 = 12 cm²

Method 2:

This method also applies the same logic but we use a direct formula to find the area of a rectangle.

• For example, if the length of the rectangle is missing, and we know the diagonal and width, then we can express the length in terms of the width and the diagonal and then use it in the formula of the area of a rectangle.
• We know that (Diagonal)² = (Length)² + (Width)². So this can also be written as, (Length)² = (Diagonal)² - (Width)². Further, this can be expressed as, Length = ⎷[(Diagonal)² - (Width)²]. Here, the length is expressed in terms of the diagonal and width. This formula can be used in place of the length.
• We know that the formula to calculate the area of a rectangle is, Area of rectangle = Length × Width. Now, if we replace the above formula of length in the area of rectangle formula, then, the area of the rectangle formula can be expressed as follows, Area of rectangle = ⎷[(Diagonal)² - (Width)²] × Width
• So, Area of a Rectangle = Width ⎷[(Diagonal)² - (Width)²]
• We can use the same formula if we know the length and if the width is missing.

Let us use an example to understand this.

Example: Find the area of a rectangle in which the width is 6 units, and the diagonal is 10 units.

Solution: We know the diagonal and the width of the rectangle. So, let us apply the direct formula, Area of a Rectangle = Width ⎷[(Diagonal)² - (Width)²]

After substituting the values of diagonal = 10 units, and width = 6 units, we get, Area of a Rectangle = Width ⎷[(Diagonal)² - (Width)²]

Area = 6 ⎷[(10)² - (6)²]

Area = 6 ⎷(100 - 36)

Area = 6 × ⎷64

Area of the rectangle = 48 square units

Area of Rectangle using Perimeter

The area of a rectangle can be calculated if the perimeter and one of its sides is given. With the help of the perimeter, we can find the unknown side and then calculate the area using the same formula, Area of rectangle = Length × Width. Let us understand this with the help of an example.

Example: Find the area of a rectangle if the perimeter is 24 units and the length is 7 units.

Solution: Perimeter of the rectangle = 24 units, length = 7 units. We can find the width using the formula,

Perimeter of rectangle = 2 (l + w)

24 = 2 (7 + w)

12 = 7 + w

w = 5 units

Now that we know the width = 5 units and the length = 7 units, we can find the area.

Area of a rectangle = l × w

Area = 7 × 5 = 35 square units

The formula of the rear of a rectangle can be derived using the following explanation. Rectangle ABCD has a diagonal. We can see that the diagonal AC divides the rectangle ABCD into two congruent triangles. The area of the rectangle is the sum of the area of these two triangles.

Area of Rectangle ABCD = Area of Triangle ABC + Area of Triangle ADC

Since both the triangles (ABC and ADC) are congruent, we can express it as follows.

Area of Rectangle ABCD = 2 × Area of Triangle ABC

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

Area of Rectangle ABCD = 2 × 1/2 × AB × BC

Area of Rectangle ABCD = AB × BC

Therefore, Area of Rectangle = Length × Width

• Area of Triangle
• Area of square
• Area of Circle
• Perimeter of Rectangle
• Difference Between Area and Perimeter

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What is the Area of a Rectangle in Geometry?

The area of a rectangle is the space occupied within the boundary of the rectangle. It is calculated by finding the product of the length and breadth (width) of the rectangle and is expressed in square units.

What is the Perimeter and Area of a Rectangle?

The perimeter of a rectangle is the sum of its four sides. It is calculated using the formula, Perimeter of Rectangle = 2 (Length + Width). The area of a rectangle is the region covered within its boundary and this can be calculated using the formula, area of rectangle = length × width.

What is the Formula of Area of Rectangle?

The formula for the area of a rectangle is the product of its length and width (breadth). This is expressed as, Area of Rectangle = length × width

What is the Unit of Area of a Rectangle?

The area of a rectangle is expressed in square units. For example, if the dimensions of a rectangle are given in inches, then the area is expressed in square inches (in²). Similarly, if the dimensions are given in cm, then the area is expressed in square cm (cm²). For example, if the length of a rectangle is 4 cm, and its width is 3 cm then its area = 4 × 3 = 12 cm²

How to Find the Area of a Rectangle Using Diagonal?

The area of a rectangle can be calculated if the diagonal and one of its sides is given. We can find the value of the missing side using the Pythagoras theorem and then find the area. For example, let us find the area of a rectangle in which the diagonal is 10 units and its length is 8 units using the following steps.

• Step 1: In this case, we can find the width using the formula, Width = ⎷[(Diagonal)² - (Length)²]
• Step 2: After substituting the given values, we get, width = ⎷[(10)² - (8)²] = ⎷36 = 6 units
• Step 3: Now, we know that the length = 8 units, width = 6 cm. So, the area of the rectangle = l × w. In this case, Area = 8 × 6 = 48 unit²

What is a Rectangle?

A rectangle is a closed two-dimensional figure with four sides where opposite sides are equal and parallel to each other. The rectangle shape has all the angles equal to 90°.

How to Find the Area of a Rectangle with Fractions?

The area of a rectangle with fractions can be calculated in the same way as we do for the other whole numbers. We just need to simplify fractions using their rules of simplification. If we have proper fractions, we can simply use the formula of the area of a rectangle and multiply the length and width, However, if we have the dimensions given in mixed fractions, we need to convert them to improper fractions and then simplify using the same formula. For example, let us find the area of a rectangle in which the length is given as \(2\dfrac{1}{3}\) units and width is given as \(1\dfrac{1}{3}\) units, now we will convert the mixed fractions to improper fractions and we get the length as 7/3, and the width as, 4/3. Now we can find the area, Area of rectangle = 7/3 × 4/3 = 28/9 = \(3\dfrac{1}{9}\) square units.

How to Find the Area of a Rectangle when Perimeter is Given?

The area of a rectangle can be calculated if the perimeter and one of its sides is given. With the help of the perimeter, we can find the missing side and then calculate the area of a rectangle using the same formula, Area of rectangle = Length × Width. Let us understand this with the help of an example. For example, let us find the area of a rectangle whose perimeter is 30 units and width is 4 units.

• Step 1: We can find the length using the perimeter and width with the help of the formula, perimeter = 2(l + w). After substituting the values we get, 30 = 2(l + 4). And the value of 'l' is 11 units.
• Step 2: Now, we have the length = 11 units, width = 4 units. So, we can find the area using the formula, Area of rectangle = l × w
• Step 3: After substituting the values in the formula, we get, Area of rectangle = l × w = 11 × 4 = 44 square units