To find the radius of the sector given the area and the angle subtended at the center, you can use the formula for the area of a sector:
\[ \text{Area of the sector} = \frac{\theta}{360} \times \pi r^2 \]
Where:
- \(\theta\) is the angle in degrees,
- \(r\) is the radius of the sector.
Given:
- Area of the sector = 22 cm²
- \(\theta = 140\) degrees
Now, we can rearrange the formula to solve for \(r\):
- Substitute the values into the formula:
\[ 22 = \frac{140}{360} \times \pi r^2 \]
- Simplify \(\frac{140}{360}\):
\[ \frac{140}{360} = \frac{7}{18} \]
Now we have:
\[ 22 = \frac{7}{18} \times \pi r^2 \]
- To isolate \(r^2\), multiply both sides by \(\frac{18}{7}\):
\[ 22 \times \frac{18}{7} = \pi r^2 \]
- Calculate the left side:
\[ \frac{22 \times 18}{7} = \frac{396}{7} \approx 56.5714 \]
So:
\[ \pi r^2 = \frac{396}{7} \]
- Now, divide both sides by \(\pi\) to solve for \(r^2\):
\[ r^2 = \frac{396}{7\pi} \]
- Finally, take the square root to find \(r\):
\[ r = \sqrt{\frac{396}{7\pi}} \]
Using \(\pi \approx 3.14159\):
\[ r \approx \sqrt{\frac{396}{7 \times 3.14159}} \approx \sqrt{\frac{396}{21.99193}} \approx \sqrt{18.0009} \approx 4.24 \text{ cm} \]
Thus, the radius of the sector is approximately 4.24 cm.