Recall the Formula for the Area of a Sector

Oct 13, 2025 by windailys.com 136 views

Introduction

In geometry, a sector of a circle is a region bounded by two radii and an arc. The area of a sector is a fraction of the total area of the circle, and it is determined by the central angle subtended by the arc. In this article, we will discuss how to find the radius of a circle sector given its area and central angle.

Understanding the Problem

The problem states that a sector of a circle has an area of 40π square units and a central angle of 75 degrees. We need to find the radius of the circle.

Recall the Formula for the Area of a Sector

The area of a sector is given by the formula:

A = (θ/360) × πr^2

where A is the area of the sector, θ is the central angle in degrees, and r is the radius of the circle.

Given Values

Area of the sector (A) = 40π square units
Central angle (θ) = 75 degrees

Substitute the Given Values into the Formula

Substituting the given values into the formula, we get:

40π = (75/360) × πr^2

Simplify the Equation

To simplify the equation, we can cancel out the π term on both sides:

40 = (75/360) × r^2

Multiply Both Sides by 360

Multiplying both sides by 360, we get:

14400 = 75r^2

Divide Both Sides by 75

Dividing both sides by 75, we get:

192 = r^2

Take the Square Root of Both Sides

Taking the square root of both sides, we get:

r = √192

Simplify the Square Root

Simplifying the square root, we get:

r = √(64 × 3) = √64 × √3 = 8√3

Conclusion

Therefore, the radius of the circle sector is 8√3 units.

Key Takeaways

The area of a sector is a fraction of the total area of the circle, determined by the central angle.
The formula for the area of a sector is A = (θ/360) × πr^2.
To find the radius of a circle sector, we can substitute the given values into the formula and solve for r.

Practice Problems

A sector of a circle has an area of 50π square units and a central angle of 90 degrees. Find the radius of the circle.
A sector of a circle has an area of 30π square units and a central angle of 60 degrees. Find the radius of the circle.

Answer Key

r = 10 units
r = 6 units
Frequently Asked Questions (FAQs) about Circle Sectors =====================================================

Q: What is a circle sector?

A: A circle sector is a region bounded by two radii and an arc. It is a part of a circle that is enclosed by two radii and an arc.

Q: How is the area of a sector related to the total area of the circle?

A: The area of a sector is a fraction of the total area of the circle, determined by the central angle. The formula for the area of a sector is A = (θ/360) × πr^2, where A is the area of the sector, θ is the central angle in degrees, and r is the radius of the circle.

Q: What is the central angle?

A: The central angle is the angle formed by two radii of a circle. It is measured in degrees and is used to determine the area of a sector.

Q: How do I find the radius of a circle sector?

A: To find the radius of a circle sector, you can use the formula A = (θ/360) × πr^2, where A is the area of the sector, θ is the central angle in degrees, and r is the radius of the circle. You can substitute the given values into the formula and solve for r.

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

A: The formula for the area of a sector is A = (θ/360) × πr^2, where A is the area of the sector, θ is the central angle in degrees, and r is the radius of the circle.

Q: Can I find the radius of a circle sector if I know the area and the central angle?

A: Yes, you can find the radius of a circle sector if you know the area and the central angle. You can use the formula A = (θ/360) × πr^2 to solve for r.

Q: What if I have a sector with a central angle of 180 degrees?

A: If you have a sector with a central angle of 180 degrees, it means that the sector is a semicircle. The area of a semicircle is half the area of the circle, which is given by the formula A = (1/2) × πr^2.

Q: Can I find the area of a sector if I know the radius and the central angle?

A: Yes, you can find the area of a sector if you know the radius and the central angle. You can use the formula A = (θ/360) × πr^2 to find the area.

Q: What if I have a sector with a central angle of 360 degrees?

A: If you have a sector with a central angle of 360 degrees, it means that the sector is the entire circle. The area of the entire circle is given by the formula A = πr^2.

Q: Can I find the radius of a circle sector if I know the area and the central angle, but the central angle is in radians?

A: Yes, you can find the radius of a circle sector if you know the area and the central angle in radians. You can use the formula A = (θ/2π) × πr^2 to solve for r.

Q: What if I have a sector with a central angle of 90 degrees and an area of 50π square units?

A: If you have a sector with a central angle of 90 degrees and an area of 50π square units, you can use the formula A = (θ/360) × πr^2 to find the radius. Substituting the given values, you get:

50π = (90/360) × πr^2

Simplifying the equation, you get:

50 = (1/4) × r^2

Multiplying both sides by 4, you get:

200 = r^2

Taking the square root of both sides, you get:

r = √200 = √(100 × 2) = 10√2

Therefore, the radius of the circle sector is 10√2 units.

Conclusion

In this article, we have discussed some frequently asked questions about circle sectors. We have covered topics such as the area of a sector, the central angle, and how to find the radius of a circle sector. We have also provided examples and practice problems to help you understand the concepts better.