Question
Circle B is shown in the diagram. The shaded sector has an area of 3𝜋 square meters.
What is the radius of circle B?
What is the radius of circle B?
Answers
Answered by
GPT-4o mini
To find the radius of circle B when given the area of the shaded sector, we can use the formula for the area of a sector of a circle:
\[
\text{Area of sector} = \frac{\theta}{360} \times \pi r^2
\]
where:
- \(\theta\) is the central angle of the sector in degrees,
- \(r\) is the radius of the circle.
Given that the area of the shaded sector is \(3\pi\) square meters, we can set up the equation:
\[
\frac{\theta}{360} \times \pi r^2 = 3\pi
\]
To solve for \(r\), we can first divide both sides of the equation by \(\pi\):
\[
\frac{\theta}{360} \times r^2 = 3
\]
Next, multiply both sides by \(360\) to isolate \(r^2\):
\[
\theta \times r^2 = 3 \times 360
\]
\[
\theta \times r^2 = 1080
\]
Now, divide both sides by \(\theta\) (assuming \(\theta \neq 0\)):
\[
r^2 = \frac{1080}{\theta}
\]
Finally, take the square root of both sides to find \(r\):
\[
r = \sqrt{\frac{1080}{\theta}}
\]
To determine the radius \(r\), we need to know the central angle \(\theta\) of the shaded sector. If \(\theta\) is provided in your original question or context, you can substitute that value into the formula to find \(r\).
If no specific angle is given, then it cannot be calculated without additional information. If you have the value of \(\theta\), please provide it!
\[
\text{Area of sector} = \frac{\theta}{360} \times \pi r^2
\]
where:
- \(\theta\) is the central angle of the sector in degrees,
- \(r\) is the radius of the circle.
Given that the area of the shaded sector is \(3\pi\) square meters, we can set up the equation:
\[
\frac{\theta}{360} \times \pi r^2 = 3\pi
\]
To solve for \(r\), we can first divide both sides of the equation by \(\pi\):
\[
\frac{\theta}{360} \times r^2 = 3
\]
Next, multiply both sides by \(360\) to isolate \(r^2\):
\[
\theta \times r^2 = 3 \times 360
\]
\[
\theta \times r^2 = 1080
\]
Now, divide both sides by \(\theta\) (assuming \(\theta \neq 0\)):
\[
r^2 = \frac{1080}{\theta}
\]
Finally, take the square root of both sides to find \(r\):
\[
r = \sqrt{\frac{1080}{\theta}}
\]
To determine the radius \(r\), we need to know the central angle \(\theta\) of the shaded sector. If \(\theta\) is provided in your original question or context, you can substitute that value into the formula to find \(r\).
If no specific angle is given, then it cannot be calculated without additional information. If you have the value of \(\theta\), please provide it!