Certainly! To find the measure of the central angle that forms the sector, we'll use the formula for the area of a sector of a circle:
The formula for the area (A) of a sector of a circle with a central angle (θ) in degrees and radius (r) is given by:
[tex]\[ A = \left(\frac{θ}{360}\right) π r^2 \][/tex]
We are given the following:
- The area of the sector (A) is 120 square centimeters.
- The radius of the circle (r) is 20 centimeters.
We need to find the central angle (θ).
Using the formula, we can rearrange it to solve for θ:
[tex]\[ θ = \left(\frac{A \times 360}{π \times r^2}\right) \][/tex]
Now we plug in the given values:
[tex]\[ θ = \left(\frac{120 \times 360}{π \times 20^2}\right) \][/tex]
[tex]\[ θ = \left(\frac{120 \times 360}{π \times 400}\right) \][/tex]
[tex]\[ θ = \left(\frac{43200}{π \times 400}\right) \][/tex]
[tex]\[ θ = \left(\frac{43200}{1256.6371}\right) \][/tex] (Here, π is approximately 3.14159)
After performing the calculation, we find that the central angle θ is approximately:
[tex]\[ θ ≈ 34.377 \text{ degrees} \][/tex]
So, the measure of the central angle that forms the sector is approximately 34.377 degrees.
