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÷6radians ?

Answer 1

[AI-generated answer — Bot GPT 3.5]

First, we need to find the area of the circle with diameter 14 cm.

Radius (r) = 14 cm ÷ 2 = 7 cm

Area of the circle = πr² = π(7)² = 49π cm²

Next, we need to find the area of the minor segment subtended by a central angle of 5π/6 radians.

First, we find the area of the sector formed by the central angle:

Area of sector = (θ/2π) * Area of circle
= (5π/6)/(2π) * 49π
= (5/6) * (49)
= 40.83 cm²

Next, we need to find the area of the triangle formed by radii and sector:

We can calculate the height of the triangle using sin(θ) = h/r
h = r * sin(θ)
h = 7 * sin(5π/6)
h = 7 * (sqrt(3)/2) = 7 * (1.732/2) = 6.06 cm

Area of triangle = 0.5 * base * height
= 0.5 * 7 * 6.06
= 21.19 cm²

Finally, the area of the minor segment can be found by subtracting the area of the triangle from the area of the sector:

Area of minor segment = Area of sector - Area of triangle
= 40.83 - 21.19
≈ 19.64 cm²

Therefore, the exact area of the major segment in which the minor segment subtends a central angle of 5π/6 radians is approximately 19.64 cm².