Answer:
- central angle: AFB: 2π/3, BFC: π/4, CFD: π/6, AFE: π/3
- sector area: AFB: 3π, BFC: 9π/8, CFD: 3π/4, AFE: 3π/2
Explanation:
You want the central angle in radians and the corresponding sector area in a circle of diameter 6 units.
Central angle
For the purpose of finding arc length and/or sector or segment area in a circle, it is useful to have the central angle expressed in radians. Just as there are 360° in a full circle, so there are 2π radians in a full circle. This means each degree is (2π)/(360) radians, or π/180 radians.
To find the radian measure of an angle given in degrees, multiply its degree measure by π/180. (The attached calculator does that with its DegreeToRadian function.)
The radian measures of the central angles in the figure are ...
- ∠AFB = 120° = 2π/3 radians
- ∠BFC = 45° = π/4 radians
- ∠CFD = 30° = π/6 radians
- ∠AFE = 60° = π/3 radians
Sector area
The area of a sector is given by ...
A = 1/2r²θ . . . . where r is the radius, and θ is the central angle in radians
The circle has a diameter of 6 units. The radius is half that, or 3 units. For each central angle θ, the corresponding area will be ...
A = (1/2)(3²)θ = (9/2)θ
For the above central angles (θ), the corresponding sector areas are ...
- AFB = (9/2)(2π/3) = 3π . . . . square units
- BFC = (9/2)(π/4) = 9π/8
- CFD = (9/2)(π/6) = 3π/4
- AFE = (9/2)(π/3) = 3π/2
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Additional comment
The attached calculator screen shows the angle measures and the areas for the sectors listed.
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