Question

What is the exact area of the major segment of a circle of radius 7cm 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 of 5π/6 radians and then subtract the area of the triangle formed by the radii and chord of the minor segment.

Area of sector = (θ/2) * r^2
= (5π/6) * 7^2
= (5π/6) * 49
= 35π/2

To find the area of the triangle formed by the radii and the chord of the minor segment, we need to find the height of the triangle. The height is given by:

h = r - rcos(θ/2)
= 7 - 7cos(5π/12)
= 7 - 7cos(150°)
= 7 - 7(-√3/2)
= 7 + (7√3)/2
= 7(2 + √3)/2
= 7 + 7√3 / 2

The area of the triangle is given by:

Area of triangle = 1/2 * base * height
= 1/2 * 2r * h
= 1/2 * 2 * 7 * (7 + 7√3 / 2)
= 7(7 + 7√3)
= 49 + 49√3

Therefore, the area of the major segment of the circle is:

Area of major segment = Area of sector - Area of triangle
= 35π/2 - (49 + 49√3)
= 17.5π - 49 - 49√3
≈ 34.48 cm^2

So, the exact area of the major segment of the circle is 17.5π - 49 - 49√3 square cm.