Answer:
Therefore, the area of sector ABC is approximately 6.60 square units.
Step-by-step explanation:
Calculate the central angle: We are given that m∠ABC = 66°. This is the measure of the central angle that intercepts sector ABC.
Find the area of the whole circle: Since we don't have the circle's radius, we can't directly calculate the area of the sector. However, we can express it as a fraction of the whole circle's area.
The ratio between the sector's central angle (θ) and 360° (full circle) is equal to the ratio between the sector's area (As) and the whole circle's area (Ac).
We can express this mathematically as:
θ / 360° = As / Ac
Substitute known values:
θ = 66° (from the given information)
We don't have Ac (area of the whole circle) yet, but we can represent it as Ac = πr² (where r is the circle's radius).
Solve for the ratio of sector area to circle area:
(66°) / 360° = As / (πr²)
Area of sector ABC relative to circle B:
As / Ac = (66° / 360°) ≈ 0.1833 (round to four decimal places)
Relate sector area to circle area using AB:
We are given that AB = 12. This represents the diameter of the circle (distance across the circle passing through the center). The radius (r) is half the diameter, so r = AB / 2 = 12 / 2 = 6.
Now we can substitute the value of r back into the expression for the whole circle's area (Ac):
Ac = πr² = π * (6²) = 36π
Calculate the actual area of sector ABC:
Since we know the ratio of the sector area to the whole circle's area (As / Ac) and the whole circle's area (Ac), we can find the actual area of sector ABC (As):
As = (As / Ac) * Ac = 0.1833 * 36π ≈ 6.60 (round to the nearest hundredth)
Therefore, the area of sector ABC is approximately 6.60 square units.