We can find the radius of the circle using the formula for the area of a sector. The area $A$ of a sector is given by the formula:

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

Where:

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

From the problem:

• The area of the sector $A = 12.36$ square meters,
• The central angle $\theta = 89^\circ$.

Substitute these values into the formula:

$12.36 = \frac{89^\circ}{360^\circ} \times \pi r^2$

Now, solve for $r$:

1. Multiply both sides by $360^\circ$ to eliminate the denominator:

$12.36 \times 360 = 89^\circ \times \pi r^2$

2. Simplify:

$4449.6 = 89^\circ \times \pi r^2$

3. Divide both sides by $89^\circ \pi$:

$r^2 = \frac{4449.6}{89^\circ \times \pi}$

4. Calculate the denominator $89^\circ \times \pi$:

$89^\circ \times \pi \approx 89 \times 3.1416 = 279.064$

$r^2 = \frac{4449.6}{279.064} \approx 15.94$

5. Finally, take the square root to find $r$:

$r \approx \sqrt{15.94} \approx 3.99 \, \text{meters}$

So, the radius of the circle is approximately 3.99 meters.

Would you like further clarification or details on the solution? Here are some related questions you might find helpful:

1. How would the radius change if the angle were different?
2. What is the formula for the area of a sector of a circle?
3. How do you find the area of the sector when you know the arc length instead of the central angle?
4. What other methods can be used to calculate the radius of a circle if the area of the sector is given?
5. How does the size of the central angle affect the area of the sector?

Tip: When dealing with sectors and angles, always make sure that your angle is in the correct units (degrees or radians) and that you use the correct formula for the area.