Answered

A sector in a circle with radius of 20 centimeters has an area of 120 square centimeters. What is the measure of the central angle that forms the
sector? Explain or show your reasoning.



Answer :

GinnyAnswer
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.

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