Determining the Area of a Rectangle: Using Perimeter and One Side

Understanding how to calculate the area of a rectangle when given the perimeter and the length of one side is crucial in many real-world applications. This guide will walk you through the process with clear, step-by-step instructions and practical examples.

Understanding the Formulas

First, let's introduce the formulas that will help us in this process:

The perimeter P of a rectangle is given by the formula P 2L 2W, where L is the length and W is the width. The area A of a rectangle is calculated using A L × W.

Rearranging the Perimeter Formula

When given the perimeter and the length of one side, we can rearrange the perimeter formula to find the width:

Given:

Perimeter P Length L Substitute L into the perimeter formula to solve for W: [ W frac{P}{2} - L ]

Calculating the Area

Once we have the width W, we can easily calculate the area:

Area A L × W

Example 1: Given a Perimeter and One Side

Suppose the perimeter P is 30 units and the length L is 8 units:

Calculate the width W: [ W frac{30}{2} - 8 15 - 8 7 text{ units} ] Calculate the area A: [ A 8 × 7 56 text{ square units} ]

The area of the rectangle is 56 square units.

Using Quadratic Equations

In cases where the length and width are not directly given, we can use the quadratic equation to find the dimensions. For instance:

L12, W5

The area A of the rectangle is 60 cm and the width W is 1 cm less than half of the length L. The equation is:

[ A W × L W × frac{L}{2} - 1 Rightarrow 60 W × 12 - 1 ]

After some manipulation, we get the quadratic equation:

[ W^2 - W - 30 0 ]

Applying the quadratic formula:

[ W_{1,2} frac{-b pm sqrt{b^2 - 4ac}}{2a} ]

In this case:

a 1 b -1 c -30

So:

[ W_{1,2} frac{1 pm sqrt{1 120}}{2} frac{1 pm 11}{2} ]

This gives us two solutions:

[ W_1 6 text{ cm and } W_2 -5 text{ cm} ]

Since width cannot be negative, W 6 cm. The length L is then 12 cm, and the perimeter is:

[ P 2 × (6 12) 36 text{ cm} ]

The diagonal d is:

[ d sqrt{6^2 12^2} sqrt{180} 6sqrt{5} text{ cm} approx 13.4 text{ cm} ]

Another Example: Given Perimeter and One Side

If the perimeter P is 35 m and the length L is 12 m:

Calculate the width W: [ W frac{35}{2} - 12 17.5 - 12 5.5 text{ m} ] Calculate the area A: [ A 12 × 5.5 66 text{ square meters} ]

The area of the rectangle is 66 square meters.

Conclusion

Knowing how to use the perimeter and one side of a rectangle to find the area can be a practical skill in various fields, from architecture to engineering. By following these steps and using the formulas provided, you can accurately determine the area of a rectangle when given only these parameters.