The Area Of The Sector Of A Circle With Radius 7 Cm Is 22 Cm² Find The Area Of The Remaining Part Of The Circle.

Mar 13, 2025 by ADMIN 114 views

The Area of the Sector of a Circle with Radius 7 cm is 22 cm²: Find the Area of the Remaining Part of the Circle

Introduction

In geometry, a circle is a set of points that are all equidistant from a central point called the center. The area of a circle is a fundamental concept in mathematics, and it is calculated using the formula A = πr², where A is the area and r is the radius of the circle. However, when a sector of a circle is removed, the remaining part of the circle is also of interest. In this article, we will explore how to find the area of the remaining part of a circle when the area of the sector is given.

The Area of a Sector of a Circle

The area of a sector of a circle is a portion of the total area of the circle. It is calculated using the formula A = (θ/360) × πr², where A is the area of the sector, θ is the central angle of the sector in degrees, and r is the radius of the circle. In this problem, we are given that the area of the sector is 22 cm² and the radius of the circle is 7 cm.

Calculating the Central Angle of the Sector

To find the area of the remaining part of the circle, we need to first find the central angle of the sector. We can do this by rearranging the formula for the area of a sector to solve for θ:

θ = (A × 360) / (πr²)

Substituting the given values, we get:

θ = (22 × 360) / (π × 7²) θ = 7920 / (3.14 × 49) θ = 7920 / 153.86 θ ≈ 51.4°

Calculating the Area of the Remaining Part of the Circle

Now that we have the central angle of the sector, we can find the area of the remaining part of the circle. The area of the remaining part is equal to the total area of the circle minus the area of the sector. The total area of the circle is given by the formula A = πr², where A is the area and r is the radius of the circle.

Substituting the given values, we get:

A = π × 7² A = 3.14 × 49 A ≈ 153.86 cm²

The area of the sector is given as 22 cm². Therefore, the area of the remaining part of the circle is:

Area of remaining part = Total area of circle - Area of sector = 153.86 cm² - 22 cm² = 131.86 cm²

Conclusion

In this article, we have explored how to find the area of the remaining part of a circle when the area of the sector is given. We have used the formula for the area of a sector to find the central angle of the sector, and then used the formula for the total area of the circle to find the area of the remaining part. The area of the remaining part of the circle is approximately 131.86 cm².

Formula Summary

Area of a sector: A = (θ/360) × πr²
Central angle of a sector: θ = (A × 360) / (πr²)
Total area of a circle: A = πr²

Example Problems

Find the area of the remaining part of a circle with a radius of 10 cm and a sector area of 30 cm².
Find the central angle of a sector with a radius of 5 cm and an area of 15 cm².
Find the area of a circle with a radius of 8 cm.

Step-by-Step Solutions

Step 1: Find the central angle of the sector using the formula θ = (A × 360) / (πr²).
Step 2: Find the area of the remaining part of the circle using the formula Area of remaining part = Total area of circle - Area of sector.
Step 3: Substitute the given values into the formulas and solve for the unknown quantities.

Real-World Applications

The area of a sector of a circle is used in various real-world applications, such as:

Calculating the area of a circular region on a map.
Finding the area of a circular cross-section of a pipe.
Determining the area of a circular region on a graph.

Tips and Tricks

When working with sectors of circles, it is essential to remember that the area of a sector is a portion of the total area of the circle.
To find the area of the remaining part of the circle, you need to first find the central angle of the sector.
Use the formula for the total area of the circle to find the area of the remaining part.

Final Thoughts

In conclusion, finding the area of the remaining part of a circle when the area of the sector is given requires a clear understanding of the formulas for the area of a sector and the total area of a circle. By following the steps outlined in this article, you can easily find the area of the remaining part of a circle.

Introduction

In our previous article, we explored how to find the area of the remaining part of a circle when the area of the sector is given. We used the formula for the area of a sector and the total area of a circle to find the area of the remaining part. In this article, we will answer some frequently asked questions related to the area of a sector of a circle.

Q&A

Q1: What is the formula for the area of a sector of a circle?

A1: The formula for the area of a sector of a circle is A = (θ/360) × πr², where A is the area of the sector, θ is the central angle of the sector in degrees, and r is the radius of the circle.

Q2: How do I find the central angle of a sector?

A2: To find the central angle of a sector, you can use the formula θ = (A × 360) / (πr²), where A is the area of the sector, and r is the radius of the circle.

Q3: What is the total area of a circle?

A3: The total area of a circle is given by the formula A = πr², where A is the area and r is the radius of the circle.

Q4: How do I find the area of the remaining part of a circle?

A4: To find the area of the remaining part of a circle, you need to first find the central angle of the sector using the formula θ = (A × 360) / (πr²). Then, you can find the area of the remaining part using the formula Area of remaining part = Total area of circle - Area of sector.

Q5: What are some real-world applications of the area of a sector of a circle?

A5: The area of a sector of a circle is used in various real-world applications, such as:

Calculating the area of a circular region on a map.
Finding the area of a circular cross-section of a pipe.
Determining the area of a circular region on a graph.

Q6: How do I calculate the area of a sector of a circle with a given radius and central angle?

A6: To calculate the area of a sector of a circle with a given radius and central angle, you can use the formula A = (θ/360) × πr², where A is the area of the sector, θ is the central angle of the sector in degrees, and r is the radius of the circle.

Q7: What is the relationship between the area of a sector and the total area of a circle?

A7: The area of a sector is a portion of the total area of the circle. The area of the sector is given by the formula A = (θ/360) × πr², where A is the area of the sector, θ is the central angle of the sector in degrees, and r is the radius of the circle.

Q8: How do I find the radius of a circle given the area of a sector and the central angle?

A8: To find the radius of a circle given the area of a sector and the central angle, you can use the formula r = (A × 360) / (θ × π), where A is the area of the sector, θ is the central angle of the sector in degrees, and r is the radius of the circle.