Question

What is the exact area of the major segment of a circle of diameter 14 cm in which the Minor segment subtends a central angle of 5π÷6 radians?

Answer 1

[AI-generated answer — Bot GPT 3.5]

To find the area of the major segment of a circle, we first need to find the area of the sector formed by the central angle and then subtract the area of the minor segment.

The area of the sector formed by the central angle of 5π/6 radians can be calculated using the formula:

Area of sector = (θ/2π) x πr^2
Area of sector = (5π/6 * 1/2π) * π(14/2)^2
Area of sector = (5/12) * 154
Area of sector = 64.1667 cm²

Next, we need to find the area of the minor segment. The minor segment is the sector minus the triangle with the central angle as one of its angles. The area of the minor segment can be calculated using the formula:

Area of minor segment = Area of sector - Area of triangle

The area of the triangle can be calculated using the formula:

Area of triangle = 1/2 * r^2 * sin(θ)
Area of triangle = 1/2 * (14/2)^2 * sin(5π/6)
Area of triangle = 21 * sqrt(3)
Area of triangle = 36.373 cm²

Area of minor segment = 64.1667 - 36.373
Area of minor segment = 27.794 cm²

Finally, the area of the major segment would be the area of the sector minus the area of the minor segment:

Area of major segment = Area of sector - Area of minor segment
Area of major segment = 64.1667 - 27.794
Area of major segment = 36.3727 cm²

Therefore, the exact area of the major segment of a circle with a diameter of 14 cm and with a minor segment subtending a central angle of 5π/6 radians is 36.3727 cm².