Calculating the Area of a Rectangle with a Given Perimeter and Ratio

In many mathematical contexts, especially in geometry, it's often required to find the dimensions and area of a rectangle given certain constraints. This article explores how to calculate the area of a rectangle when the ratio of the length and breadth is given, alongside the perimeter.

Understanding the Given Information

Suppose the perimeter of a rectangle is 20 cm and the ratio of the length to the breadth is 3:2. We are tasked with finding the area of this rectangle. Let's walk through the steps to solve this problem systematically.

Step 1: Expressing Length and Breadth Using a Variable

Given the ratio of length and breadth as 3:2, let the length be 3x and the breadth be 2x, where x is a common factor.

Step 2: Using the Perimeter Formula to Find the Value of x

The formula to calculate the perimeter of a rectangle is:

P 2(length breadth)

Substituting the lengths and breadths into the formula, we get:

20 2(3x 2x)

Simplifying the equation:

20 2(5x) 1

Dividing both sides by 10, we find:

x 2

Step 3: Calculating the Dimensions

Using the value of x 2, we can find the actual dimensions:

Length 3x 3(2) 6 cm Breadth 2x 2(2) 4 cm

Step 4: Calculating the Area

The area of a rectangle is calculated using the formula:

A length × breadth

Substituting the dimensions:

A 6 cm × 4 cm 24 cm2

Alternative Scenarios and Calculations

Let's explore a slightly different scenario where the perimeter is 30 cm instead.

Scenario with Different Perimeter

Assume the width is 2 cm and the length is 3 cm. However, the actual perimeter is 30 cm, which is three times greater than 10 cm (the perimeter for the smaller rectangle). Consequently, the actual width and length are 6 cm and 9 cm respectively.

The area of the rectangle can now be calculated as:

A width × length 6 cm × 9 cm 54 cm2

Solving with Algebraic Expressions

Using algebraic expressions, we can set up the following equations:

3x 2y

The perimeter equation is:

2x 2y 30

Solving for x and y:

3x 2y Substituting y (3x)/2 into the perimeter equation: 2x 2(3x/2) 30 2x 3x 30 5x 30 x 6 y (3(6))/2 9

The area is then:

A x × y 6 cm × 9 cm 54 cm2

Conclusion

By following these steps, we can accurately calculate the area of a rectangle given its perimeter and a specified ratio of the length to the breadth. The area of the rectangle, in both scenarios, is 54 cm2 when the length to breadth ratio is 3:2, and the perimeter is 30 cm.