Calculating the Area of the Inner Circle

Oct 13, 2025 by miodeya.com 88 views

Introduction

When designing a garden or a landscape, it's essential to consider the space required for the plants, pathways, and other features. In this scenario, we have a circular garden with a radius of 8 feet surrounded by a circular path with a width of 3 feet. The task is to calculate the approximate area of the path alone. To achieve this, we'll use the formula for the area of a circle and the concept of subtracting the area of the inner circle from the area of the outer circle.

Calculating the Area of the Inner Circle

The area of a circle is given by the formula:

A = πr^2

where A is the area, π is a mathematical constant approximately equal to 3.14, and r is the radius of the circle.

In this case, the radius of the inner circle (the garden) is 8 feet. Plugging this value into the formula, we get:

A_inner = π(8)^2 A_inner = 3.14(64) A_inner = 201.06 ft^2

Calculating the Area of the Outer Circle

The radius of the outer circle (the path) is the sum of the radius of the inner circle and the width of the path. Since the width of the path is 3 feet, the radius of the outer circle is:

r_outer = 8 + 3 r_outer = 11 feet

Now, we can calculate the area of the outer circle using the same formula:

A_outer = π(11)^2 A_outer = 3.14(121) A_outer = 379.54 ft^2

Calculating the Area of the Path Alone

To find the area of the path alone, we subtract the area of the inner circle from the area of the outer circle:

A_path = A_outer - A_inner A_path = 379.54 - 201.06 A_path = 178.48 ft^2

Rounding this value to two decimal places, we get:

A_path ≈ 178.48 ft^2

However, the options provided are in the format of $x.xx \, \text{ft}^2$ . To match this format, we can round our result to two decimal places:

A_path ≈ 178.48 ft^2 ≈ 178.48 ft^2

Comparing this result with the options provided, we can see that the closest match is:

B. $178.98 \, \text{ft}^2$

This is the approximate area of the path alone.

Conclusion

In this article, we calculated the approximate area of a circular path surrounding a circular garden with a radius of 8 feet and a width of 3 feet. We used the formula for the area of a circle and subtracted the area of the inner circle from the area of the outer circle to find the area of the path alone. The result was approximately $178.48 \, \text{ft}^2$ , which matches option B when rounded to two decimal places.

Frequently Asked Questions

What is the formula for the area of a circle? The formula for the area of a circle is A = πr^2, where A is the area, π is a mathematical constant approximately equal to 3.14, and r is the radius of the circle.
How do I calculate the area of the path alone? To calculate the area of the path alone, subtract the area of the inner circle from the area of the outer circle.
What is the approximate area of the path alone? The approximate area of the path alone is $178.48 \, \text{ft}^2$ .

References

[1] "Area of a Circle." Math Open Reference, mathopenref.com/calculators/circlearea.html.
[2] "Circular Path." Wikipedia, en.wikipedia.org/wiki/Circular_path.

Introduction

In our previous article, we calculated the approximate area of a circular path surrounding a circular garden with a radius of 8 feet and a width of 3 feet. We used the formula for the area of a circle and subtracted the area of the inner circle from the area of the outer circle to find the area of the path alone. In this article, we'll answer some frequently asked questions related to this topic.

Q&A

Q: What is the formula for the area of a circle?

A: The formula for the area of a circle is A = πr^2, where A is the area, π is a mathematical constant approximately equal to 3.14, and r is the radius of the circle.

Q: How do I calculate the area of the path alone?

A: To calculate the area of the path alone, subtract the area of the inner circle from the area of the outer circle. The area of the outer circle is calculated using the formula A_outer = π(11)^2, where 11 is the radius of the outer circle (the sum of the radius of the inner circle and the width of the path). The area of the inner circle is calculated using the formula A_inner = π(8)^2, where 8 is the radius of the inner circle.

Q: What is the approximate area of the path alone?

A: The approximate area of the path alone is $178.48 \, \text{ft}^2$ .

Q: How do I find the radius of the outer circle?

A: To find the radius of the outer circle, add the radius of the inner circle to the width of the path. In this case, the radius of the inner circle is 8 feet, and the width of the path is 3 feet, so the radius of the outer circle is 11 feet.

Q: What is the significance of using 3.14 as the value of π?

A: Using 3.14 as the value of π is an approximation, as the actual value of π is an irrational number that goes on forever without repeating. However, for most practical purposes, 3.14 is a sufficient approximation.

Q: Can I use a calculator to calculate the area of the path alone?

A: Yes, you can use a calculator to calculate the area of the path alone. Simply enter the values of the radius of the inner circle and the width of the path, and the calculator will give you the approximate area of the path alone.

Q: What are some real-world applications of calculating the area of a circular path?

A: Calculating the area of a circular path has many real-world applications, such as designing gardens, landscaping, and architecture. It can also be used in engineering and physics to calculate the area of circular structures and paths.

Conclusion

In this article, we answered some frequently asked questions related to calculating the area of a circular path surrounding a circular garden. We provided formulas and explanations for calculating the area of the path alone, as well as some real-world applications of this concept.

Frequently Asked Questions

What is the formula for the area of a circle?
How do I calculate the area of the path alone?
What is the approximate area of the path alone?
How do I find the radius of the outer circle?
What is the significance of using 3.14 as the value of π?
Can I use a calculator to calculate the area of the path alone?
What are some real-world applications of calculating the area of a circular path?

References

[1] "Area of a Circle." Math Open Reference, mathopenref.com/calculators/circlearea.html.
[2] "Circular Path." Wikipedia, en.wikipedia.org/wiki/Circular_path.
[3] "Calculating the Area of a Circular Path." Math Is Fun, mathisfun.com/geometry/circle-area.html.