A Circle Has An Area Of $30 \, \text{in}^2$. What Is The Area Of A $60^\circ$ Sector Of This Circle?A. $6 \, \text{in}^2$ B. $5 \, \text{in}^2$ C. $3 \, \text{in}^2$ D. $10 \, \text{in}^2$

Introduction

When dealing with circles, it's essential to understand the relationship between the whole circle and its parts, such as sectors. A sector is a portion of a circle enclosed by two radii and an arc. In this article, we will explore how to find the area of a sector of a circle, given the area of the entire circle. We will use the example of a circle with an area of $30 \, \text{in}^2$ and find the area of a $60^\circ$ sector.

The Formula for the Area of a Circle

The area of a circle is given by the formula:

$$A = \pi r^2$$

where $A$ is the area of the circle and $r$ is the radius. In this case, we are given that the area of the circle is $30 \, \text{in}^2$. We can use this information to find the radius of the circle.

Finding the Radius of the Circle

To find the radius of the circle, we can rearrange the formula for the area of a circle to solve for $r$:

$$r = \sqrt{\frac{A}{\pi}}$$

Plugging in the given area of $30 \, \text{in}^2$, we get:

$$r = \sqrt{\frac{30}{\pi}}$$

The Formula for the Area of a Sector

The area of a sector of a circle is given by the formula:

$$A_{\text{sector}} = \frac{\theta}{360} \pi r^2$$

where $A_{\text{sector}}$ is the area of the sector, $\theta$ is the central angle of the sector in degrees, and $r$ is the radius of the circle. In this case, we are given that the central angle of the sector is $60^\circ$.

Finding the Area of the Sector

Now that we have the radius of the circle and the central angle of the sector, we can plug these values into the formula for the area of a sector:

$$A_{\text{sector}} = \frac{60}{360} \pi \left(\sqrt{\frac{30}{\pi}}\right)^2$$

Simplifying the expression, we get:

$$A_{\text{sector}} = \frac{1}{6} \pi \left(\frac{30}{\pi}\right)$$

$$A_{\text{sector}} = 5 \, \text{in}^2$$

Conclusion

In this article, we explored how to find the area of a sector of a circle, given the area of the entire circle. We used the example of a circle with an area of $30 \, \text{in}^2$ and found the area of a $60^\circ$ sector to be $5 \, \text{in}^2$. This demonstrates the importance of understanding the relationship between the whole circle and its parts, such as sectors.

Frequently Asked Questions

• What is the area of a circle with a radius of 5 inches?
• How do I find the area of a sector of a circle?
• What is the formula for the area of a sector of a circle?

Step-by-Step Solution

1. Find the radius of the circle using the formula: $r = \sqrt{\frac{A}{\pi}}$
2. Plug the radius and central angle into the formula for the area of a sector: $A_{\text{sector}} = \frac{\theta}{360} \pi r^2$
3. Simplify the expression to find the area of the sector.

Key Takeaways

• The area of a circle is given by the formula: $A = \pi r^2$
• The area of a sector of a circle is given by the formula: $A_{\text{sector}} = \frac{\theta}{360} \pi r^2$
• To find the area of a sector, you need to know the central angle and the radius of the circle.

Real-World Applications

• Finding the area of a sector of a circle is important in various real-world applications, such as:
  • Calculating the area of a slice of pizza
  • Determining the area of a portion of a circle in a design or architecture
  • Finding the area of a sector of a circle in a scientific or engineering context

Additional Resources

• For more information on the area of a circle, see the formula: $A = \pi r^2$
• For more information on the area of a sector of a circle, see the formula: $A_{\text{sector}} = \frac{\theta}{360} \pi r^2$
• For additional practice problems and examples, see the following resources:
  • Khan Academy: Area of a circle
  • Mathway: Area of a sector of a circle
  • Wolfram Alpha: Area of a sector of a circle
    

Introduction

In our previous article, we explored how to find the area of a sector of a circle, given the area of the entire circle. We used the example of a circle with an area of $30 \, \text{in}^2$ and found the area of a $60^\circ$ sector to be $5 \, \text{in}^2$. In this article, we will answer some frequently asked questions related to the area of a circle and its sectors.

Q&A

Q: What is the area of a circle with a radius of 5 inches?

A: To find the area of a circle with a radius of 5 inches, we can use the formula: $A = \pi r^2$. Plugging in the value of the radius, we get:

$$A = \pi (5)^2$$

$$A = 25\pi \, \text{in}^2$$

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

A: To find the area of a sector of a circle, you need to know the central angle and the radius of the circle. You can use the formula: $A_{\text{sector}} = \frac{\theta}{360} \pi r^2$, where $A_{\text{sector}}$ is the area of the sector, $\theta$ is the central angle in degrees, and $r$ is the radius of the circle.

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

A: The formula for the area of a sector of a circle is: $A_{\text{sector}} = \frac{\theta}{360} \pi r^2$, where $A_{\text{sector}}$ is the area of the sector, $\theta$ is the central angle in degrees, and $r$ is the radius of the circle.

Q: How do I find the radius of a circle given its area?

A: To find the radius of a circle given its area, you can use the formula: $r = \sqrt{\frac{A}{\pi}}$, where $A$ is the area of the circle and $r$ is the radius.

Q: What is the relationship between the area of a circle and its sector?

A: The area of a sector of a circle is a portion of the area of the entire circle. The relationship between the area of a circle and its sector is given by the formula: $A_{\text{sector}} = \frac{\theta}{360} \pi r^2$, where $A_{\text{sector}}$ is the area of the sector, $\theta$ is the central angle in degrees, and $r$ is the radius of the circle.

Q: Can I find the area of a sector of a circle if I only know the area of the entire circle?

A: Yes, you can find the area of a sector of a circle if you only know the area of the entire circle. You can use the formula: $A_{\text{sector}} = \frac{\theta}{360} \pi r^2$, where $A_{\text{sector}}$ is the area of the sector, $\theta$ is the central angle in degrees, and $r$ is the radius of the circle. To find the radius, you can use the formula: $r = \sqrt{\frac{A}{\pi}}$, where $A$ is the area of the circle.

Conclusion

In this article, we answered some frequently asked questions related to the area of a circle and its sectors. We hope that this article has provided you with a better understanding of the relationship between the area of a circle and its sectors.

Additional Resources

• For more information on the area of a circle, see the formula: $A = \pi r^2$
• For more information on the area of a sector of a circle, see the formula: $A_{\text{sector}} = \frac{\theta}{360} \pi r^2$
• For additional practice problems and examples, see the following resources:
  • Khan Academy: Area of a circle
  • Mathway: Area of a sector of a circle
  • Wolfram Alpha: Area of a sector of a circle

Step-by-Step Solution

1. Find the radius of the circle using the formula: $r = \sqrt{\frac{A}{\pi}}$
2. Plug the radius and central angle into the formula for the area of a sector: $A_{\text{sector}} = \frac{\theta}{360} \pi r^2$
3. Simplify the expression to find the area of the sector.

Key Takeaways

• The area of a circle is given by the formula: $A = \pi r^2$
• The area of a sector of a circle is given by the formula: $A_{\text{sector}} = \frac{\theta}{360} \pi r^2$
• To find the area of a sector, you need to know the central angle and the radius of the circle.

Real-World Applications

• Finding the area of a sector of a circle is important in various real-world applications, such as:
  • Calculating the area of a slice of pizza
  • Determining the area of a portion of a circle in a design or architecture
  • Finding the area of a sector of a circle in a scientific or engineering context