This guide provides a step-by-step process, including formulas, explanations, and real number examples to ensure clarity and ease of understanding.
Step 1: Show the Formula
To find the radius (\( r \)) of a circular sector, we use the following formula derived from the sector area formula:
\[ r = \sqrt{\frac{2A}{\theta_{\text{radians}}}} \]
where:
- \( A \) is the area of the sector.
- \( \theta_{\text{radians}} \) is the central angle in radians.
Step 2: Explain the Formula
The formula \( r = \sqrt{\frac{2A}{\theta_{\text{radians}}}} \) is derived from the standard area formula for a circular sector \( A = \frac{1}{2} r^2 \theta_{\text{radians}} \). Rearranging this formula to solve for \( r \) gives us \( r = \sqrt{\frac{2A}{\theta_{\text{radians}}}} \).
Step 3: Convert Angle to Radians
Given that the angle is provided in degrees, we need to convert it to radians using the conversion factor \( \frac{\pi}{180^\circ} \).
Step 4: Use Actual Numbers as an Example
Let's assume the area (\( A \)) of the sector is 50 square units and the central angle (\( \theta_{\text{degrees}} \)) is 30 degrees.
Step 5: Convert the Angle to Radians
First, convert the angle from degrees to radians:
\[ \theta_{\text{radians}} = 30^\circ \times \frac{\pi}{180^\circ} = \frac{\pi}{6} \]
Step 6: Calculate the Radius
Now, substitute the given values into the formula:
\[ r = \sqrt{\frac{2 \times 50}{\frac{\pi}{6}}} \]
\[ r = \sqrt{\frac{100 \times 6}{\pi}} \]
\[ r = \sqrt{\frac{600}{\pi}} \]
Using \(\pi \approx 3.14159\):
\[ r = \sqrt{\frac{600}{3.14159}} \]
\[ r \approx \sqrt{190.985} \]
\[ r \approx 13.82 \]
Final Value
The radius of the circular sector with an area of 50 square units and a central angle of 30 degrees is approximately \( 13.82 \) units.
