To find the angle measure of an arc bounding a sector with a given area, we can use the formula for the area of a sector in a circle:
[tex]\[ A = \frac{1}{2} r^2 \theta \][/tex]
where [tex]\( A \)[/tex] is the area of the sector, [tex]\( r \)[/tex] is the radius of the circle, and [tex]\( \theta \)[/tex] is the angle in radians.
Given:
- Area of the sector, [tex]\( A = 6\pi \)[/tex] square centimeters
- Radius of the circle, [tex]\( r = 4 \)[/tex] centimeters
We need to find the angle [tex]\( \theta \)[/tex].
Step-by-step solution:
1. Start with the formula for the area of a sector:
[tex]\[ A = \frac{1}{2} r^2 \theta \][/tex]
2. Plug in the given values for [tex]\( A \)[/tex] and [tex]\( r \)[/tex]:
[tex]\[ 6\pi = \frac{1}{2} (4)^2 \theta \][/tex]
3. Simplify the expression [tex]\( (4)^2 \)[/tex]:
[tex]\[ 6\pi = \frac{1}{2} (16) \theta \][/tex]
4. Further simplify:
[tex]\[ 6\pi = 8 \theta \][/tex]
5. Solve for [tex]\( \theta \)[/tex] by dividing both sides of the equation by 8:
[tex]\[ \theta = \frac{6\pi}{8} \][/tex]
6. Simplify the fraction [tex]\( \frac{6\pi}{8} \)[/tex]:
[tex]\[ \theta = \frac{3\pi}{4} \][/tex]
Thus, the angle measure of the arc bounding the sector with an area of [tex]\( 6\pi \)[/tex] square centimeters, for a circle with a radius of 4 centimeters, is:
[tex]\[ \theta = \frac{3\pi}{4} \][/tex] radians.
