Question

Circle L is shown. Line segments K L and M L are radii. The length of K L is 12. Angle M L K is 60 degrees. Sector M L K with a 60 degree angle is shaded.
What is the area of the sector that is not shaded?

12Pi units squared
24Pi units squared
120Pi units squared
144Pi units squared

Answer 1

To find the area of the sector that is not shaded, we need to first calculate the area of the entire sector \( M L K \) and then subtract the area of the shaded portion.

1. **Calculate the area of the triangle \( M L K \):**
- The radius \( K L = 12 \).
- The area of a sector is given by the formula:
\[
\text{Area of sector} = \frac{\theta}{360} \cdot \pi r^2
\]
Where \( \theta \) is in degrees and \( r \) is the radius.

2. **Given data:**
- \( \theta = 60^\circ \)
- \( r = 12 \)

3. **Calculate the area of the sector:**
\[
\text{Area of sector} = \frac{60}{360} \cdot \pi (12^2)
\]
Simplifying:
\[
\text{Area of sector} = \frac{1}{6} \cdot \pi \cdot 144
\]
\[
\text{Area of sector} = 24\pi
\]

4. **Now, calculate the area of the circle:**
- The area of a circle is given by \( \pi r^2 \).
\[
\text{Area of circle} = \pi (12^2) = \pi \cdot 144 = 144\pi
\]

5. **Area of the sector that is not shaded:**
\[
\text{Area of the non-shaded sector} = \text{Area of the circle} - \text{Area of the shaded sector}
\]
Since \( M L K \) is a shaded sector:
\[
\text{Non-shaded area} = \text{Area of circle} - \text{Area of shaded sector}
= 144\pi - 24\pi = 120\pi
\]

Thus, the area of the sector that is not shaded is \( \boxed{120\pi} \) square units.