Calculating the Area of a Rectangle Using Its Perimeter

Introduction

Understanding the relationship between the dimensions of a rectangle and its perimeter is a fundamental concept in geometry. This article explores how to calculate the area of a rectangle when given the perimeter and the fact that its width is half the length. Let's dive into the detailed steps!

Given Conditions

We are given that the width (y) of the rectangle is half of its length (x). This relationship can be written as:

x 2y

Additionally, the perimeter of the rectangle is 24 inches. The formula for the perimeter of a rectangle is:

2(length width)

Step-by-Step Calculation

Step 1: Substitute the Relationship into the Perimeter Formula

First, let's substitute x 2y into the perimeter formula:

2(2y y) 24

Simplify the expression:

3y 12

Solve for y:

y 4

Now, find the length x, which is twice the width:

x 2(4) 8

Calculating the Area

The area of a rectangle is calculated by multiplying the length by the width:

Area length × width

Substituting the values we found for length and width:

Area 8 × 4 32 square inches

Verification

To ensure accuracy, we can check the perimeter calculation:

2(length width) 2(8 4) 2(12) 24 inches

This confirms that our solution is correct.

Conclusion

In summary, if the width of a rectangle is half of its length and the perimeter is 24 inches, the area of the rectangle is 32 square inches. This problem demonstrates the application of basic algebraic and geometric principles.

Key Concepts

Perimeter: The total distance around the edges of a shape. Area: The space inside a two-dimensional shape. Rectangle: A quadrilateral with four right angles and opposite sides of equal length.

FAQs

Q: How do you calculate the perimeter of a rectangle?

A: The perimeter of a rectangle is calculated by adding twice the length and twice the width. The formula is 2(length width).

Q: What is the area of a rectangle?

A: The area of a rectangle is calculated by multiplying its length by its width. The formula is length × width.

Q: How can you check if your answer for the area and perimeter of a rectangle is correct?

A: You can check your answer by substituting the calculated length and width back into the perimeter formula to ensure it matches the given perimeter, and by recalculating the area to confirm it is accurate.