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