How to Calculate the Perimeter of a Rectangle Given Its Area and Ratio of Length to Width

When working with geometric shapes, understanding how to calculate the perimeter based on given dimensions or relationships between them is a valuable skill. In this article, we'll walk through a detailed example of a rectangle where the length is five times its width and the area is 125 square feet. We'll then extend this concept to other scenarios involving different dimensions and areas.

Understanding the Problem: Rectangle with Given Length to Width Ratio and Area

Let's consider a rectangle where the length is five times the width. The area of this rectangle is given as 125 square feet. We need to find the perimeter of this rectangle. To solve this, we will follow these steps:

Define the variables: Use the given ratio to express one dimension in terms of the other. Apply the area formula. Calculate the dimensions of the rectangle. Use the perimeter formula to find the perimeter.

Solving the Given Example

Step 1: Define the variables

Let the width of the rectangle be w. Therefore, the length l is 5 times the width, so l 5w.

Step 2: Use the given ratio

The area of the rectangle is given as 125 square feet. The formula for the area of a rectangle is A l × w.

Substitute the given variables into the area formula:

125 (5w) × w

Step 3: Apply the area formula

Simplify the equation:

125 5w^2

Divide both sides of the equation by 5:

25 w^2

Taking the square root of both sides:

[latex]w 5 text{ ft}[/latex]

Step 4: Calculate the dimensions

Now that we have the width, we can find the length:

[latex]l 5w 5 times 5 25 text{ ft}[/latex]

The dimensions of the rectangle are: width w 5 ft and length l 25 ft.

Step 5: Use the perimeter formula

The formula for the perimeter of a rectangle is P 2l 2w.

Substitute the values of l and w into the perimeter formula:

[latex]P 2 times 25 2 times 5 50 10 60 text{ ft}[/latex]

The perimeter of the rectangle is 60 feet.

Extending to Other Examples

Let's now look at some variations of the same problem to gain a better understanding.

Example 1

Given the perimeter is 72 feet and the length is five times the width, we need to find the dimensions and the area:

Step 1: Define the variables

Let the width be w and the length l 5w.

Step 2: Use the perimeter formula

The perimeter formula is P 2l 2w. Substituting the given values:

[latex]72 2 times 5w 2w 10w 2w 12w[/latex]

Solving for w:

[latex]72 12w Rightarrow w 6 text{ ft}[/latex]

Therefore:

[latex]l 5w 5 times 6 30 text{ ft}[/latex]

Step 3: Calculate the area

The area formula for a rectangle is A l × w. Substituting the values:

[latex]A 30 times 6 180 text{ sq. ft}[/latex]

Example 2

Given 5L × 1w 180 sq. ft, find the perimeter:

Step 1: Define the variables

Let L 30 and w 6 since 5L × 1w 180 is equivalent to 30 × 6 180.

Step 2: Use the perimeter formula

The perimeter formula is P 2L 2w. Substituting the values:

[latex]P 2 times 30 2 times 6 60 12 72 text{ ft}[/latex]

Example 3

Given the length is 5 times the width and the area is 125 square feet, we need to find the dimensions and the perimeter:

Step 1: Define the variables

Let the width be w and the length l 5w.

Step 2: Apply the area formula

The area formula is A l × w. Substituting the given values:

125 (5w) × w

Solving for w and l as previously explained, we get:

[latex]w 5 text{ ft}, l 25 text{ ft}[/latex]

Step 3: Use the perimeter formula

The perimeter formula is P 2l 2w. Substituting the values:

[latex]P 2 times 25 2 times 5 50 10 60 text{ ft}[/latex]

The perimeter of the rectangle is 60 feet.

Conclusion

Understanding how to calculate the perimeter of a rectangle given its area and the ratio of its length to width is a fundamental skill in geometry. By solving these types of problems, you can develop a deeper understanding of geometric relationships and improve your problem-solving skills in mathematics. Practice with different examples can help reinforce these concepts.

Frequently Asked Questions (FAQs)

What is the perimeter of a rectangle if the length is five times the width and the area is 180 square inches?

Let's solve this step by step:

Step 1: Define the variables Step 2: Applying the area formula Step 3: Calculating the dimensions Step 4: Using the perimeter formula

After solving, the perimeter will be 72 inches.

How do I calculate the area of a rectangle if I know the length is five times the width and the perimeter is 108 inches?

Let's solve this step by step:

Step 1: Define the variables Step 2: Using the perimeter formula to find the width and length Step 3: Applying the area formula

After solving, the area will be 405 square inches.