Deriving the Perimeter of a Rectangle When Given the Side of a Square and Their Areas are Equal

Introduction

In this article, we will explore a mathematical problem involving the areas and perimeters of a square and a rectangle. We will walk through the process of solving this problem using step-by-step calculations to find the perimeter of the rectangle given the side of the square and the areas of both shapes.

Problem Statement

The perimeter of the square is 64 cm, and the areas of the square and rectangle are equal. The ratio of the length to the breadth of the rectangle is 4:1. We need to find the perimeter of the rectangle.

Step-by-Step Solution

Step 1: Calculate the Side Length of the Square

The perimeter of a square is given by the formula:

P 4s

where s is the side length of the square.

Given that the perimeter P is 64 cm, we have:

64 4s

s 64 / 4 16 , text{cm}

Step 2: Calculate the Area of the Square

The area of the square is given by the formula:

A s^2

Using the side length calculated in Step 1:

A 16^2 256 , text{sq cm}

Step 3: Set Up the Dimensions of the Rectangle

Let the length of the rectangle be 4x cm and the breadth be x cm. Given the ratio of length to breadth is 4:1, the area of the rectangle can be expressed as:

Area 4x times x 4x^2 256 , text{sq cm}

Solving for x:

x^2 256 / 4 64

x sqrt{64} 8 , text{cm}

So, the breadth of the rectangle is x 8 , text{cm}

And the length of the rectangle is:

4x 4 times 8 32 , text{cm}

Step 4: Calculate the Perimeter of the Rectangle

The perimeter of the rectangle is given by the formula:

P_r 2(l b)

where l is the length and b is the breadth of the rectangle.

Substituting the dimensions found in the previous steps:

P_r 2(32 8) 2 times 40 80 , text{cm}

Conclusion

The perimeter of the rectangle is 80 , text{cm}.