A $20^{\circ}$ Sector In A Circle Has An Area Of $21.5 \pi \text{ Yd}^2$.What Is The Area Of The Circle? Use 3.14 For $\pi$.Enter Your Answer As A Decimal In The Box.$\square$ Yd\[$^2\$\]

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Introduction

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In geometry, a sector is a part of a circle enclosed by two radii and an arc. Given the area of a sector, we can find the area of the entire circle using the ratio of the sector's angle to the full circle's angle. In this problem, we are given a $20^{\circ}$ sector in a circle with an area of $21.5 \pi \text{ yd}^2$. We will use this information to find the area of the circle.

The Formula for the Area of a Sector

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The area of a sector can be calculated using the formula:

$$A = \frac{\theta}{360} \pi r^2$$

where $A$ is the area of the sector, $\theta$ is the angle of the sector in degrees, $\pi$ is a mathematical constant approximately equal to $3.14$, and $r$ is the radius of the circle.

Given Information

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We are given that the area of the $20^{\circ}$ sector is $21.5 \pi \text{ yd}^2$. We can use this information to find the radius of the circle.

Finding the Radius of the Circle

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Using the formula for the area of a sector, we can set up the equation:

$$21.5 \pi = \frac{20}{360} \pi r^2$$

Simplifying the equation, we get:

$$21.5 \pi = \frac{1}{18} \pi r^2$$

Multiplying both sides by $18$, we get:

$$388 \pi = \pi r^2$$

Dividing both sides by $\pi$, we get:

$$388 = r^2$$

Taking the square root of both sides, we get:

$$r = \sqrt{388}$$

Using a calculator, we find that:

$$r \approx 19.73$$

Finding the Area of the Circle

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Now that we have the radius of the circle, we can find the area of the circle using the formula:

$$A = \pi r^2$$

Substituting the value of $r$, we get:

$$A = \pi (19.73)^2$$

Using a calculator, we find that:

$$A \approx 3.14 (388)$$

$$A \approx 1223.32$$

Conclusion

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In this problem, we were given a $20^{\circ}$ sector in a circle with an area of $21.5 \pi \text{ yd}^2$. We used the formula for the area of a sector to find the radius of the circle, and then used the formula for the area of a circle to find the area of the circle. We found that the area of the circle is approximately $1223.32 \text{ yd}^2$.

Final Answer

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The final answer is $\boxed{1223.32}$.

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Introduction

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In our previous article, we explored how to find the area of a circle given the area of a sector. We used the formula for the area of a sector and the formula for the area of a circle to find the radius of the circle and then the area of the circle. In this article, we will answer some common questions related to the problem.

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

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A: The formula for the area of a sector is:

$$A = \frac{\theta}{360} \pi r^2$$

where $A$ is the area of the sector, $\theta$ is the angle of the sector in degrees, $\pi$ is a mathematical constant approximately equal to $3.14$, and $r$ is the radius of the circle.

Q: How do I find the radius of the circle if I know the area of the sector?

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A: To find the radius of the circle, you can use the formula for the area of a sector and solve for $r$. Here's an example:

$$21.5 \pi = \frac{20}{360} \pi r^2$$

Simplifying the equation, we get:

$$21.5 \pi = \frac{1}{18} \pi r^2$$

Multiplying both sides by $18$, we get:

$$388 \pi = \pi r^2$$

Dividing both sides by $\pi$, we get:

$$388 = r^2$$

Taking the square root of both sides, we get:

$$r = \sqrt{388}$$

Using a calculator, we find that:

$$r \approx 19.73$$

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

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A: The formula for the area of a circle is:

$$A = \pi r^2$$

where $A$ is the area of the circle and $r$ is the radius of the circle.

Q: How do I find the area of the circle if I know the radius?

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A: To find the area of the circle, you can use the formula:

$$A = \pi r^2$$

Substituting the value of $r$, we get:

$$A = \pi (19.73)^2$$

Using a calculator, we find that:

$$A \approx 3.14 (388)$$

$$A \approx 1223.32$$

Q: What if I don't know the radius of the circle?

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A: If you don't know the radius of the circle, you can use the formula for the area of a sector to find the radius and then use the formula for the area of a circle to find the area of the circle.

Q: Can I use a calculator to find the area of the circle?

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A: Yes, you can use a calculator to find the area of the circle. Simply substitute the value of $r$ into the formula:

$$A = \pi r^2$$

and use the calculator to find the value of $A$.

Conclusion

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In this article, we answered some common questions related to finding the area of a circle given the area of a sector. We provided formulas and examples to help you understand the concepts. We hope this article has been helpful in answering your questions.

Final Answer

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The final answer is $\boxed{1223.32}$.