The given problem involves finding the radius of a circle based on the area of a sector and the central angle.

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

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

Where:

• $A$ is the area of the sector,
• $\theta$ is the central angle in degrees,
• $r$ is the radius of the circle,
• $\pi$ is approximately 3.1416.

From the problem, we know:

• The area of the sector $A = 65 \, \text{m}^2$,
• The central angle $\theta = 110^\circ$.

Step 1: Plug the known values into the formula.

$65 = \frac{110}{360} \times \pi r^2$

Step 2: Simplify the fraction.

$\frac{110}{360} = \frac{11}{36}$

Thus, the equation becomes:

$65 = \frac{11}{36} \times \pi r^2$

Step 3: Solve for $r^2$.

First, multiply both sides by $\frac{36}{11}$:

$65 \times \frac{36}{11} = \pi r^2$

$212.73 = \pi r^2$

Now, divide both sides by $\pi$:

$r^2 = \frac{212.73}{\pi} = \frac{212.73}{3.1416} = 67.69$

Step 4: Take the square root to find $r$.

$r = \sqrt{67.69} \approx 8.2 \, \text{m}$

Thus, the radius of the circle is approximately $8.2 \, \text{m}$.

Would you like further clarification or more details?

Here are 5 related questions you might explore:

1. How do we derive the formula for the area of a sector of a circle?
2. What happens to the area of a sector if the central angle is doubled but the radius remains the same?
3. Can you calculate the circumference of the full circle using this radius?
4. How would the formula change if the central angle was given in radians instead of degrees?
5. How can you apply this formula to calculate the area of a segment instead of a sector?

Tip: Always ensure the angle is in degrees (or convert to radians if needed) before applying sector-related formulas!