Answer:
9.83 mm^2
Explanation:
Radius = diameter/2 = 8/2 = 4
Area of circle: A=πr^2 = π4^2 = 16π
1/2 area of circle = 16π/2 = 8π
Since the angle = 60o, the area is 1/3 of half circle (the half circle has angle of 180o) => Area of 60o = 8π/3 = 8/3π
The last step is to find the area of the triangle.
Since the 2 sides are radius => it is isosceles triangle
the obtuse angle of the triangle = 180 - 60 = 120o
therefore the other 2 angles are 60/2 = 30o
If you draw a line from the center of the circle perpendicular to the side of the triangle (see attachment) => we have 2 equal right triangles △ACO = △BCO
so for the right triangle, radius will be the hypotenuse, the angle is 30, we need to solve for OC
sin(30o) = opposite/hypotenuse
1/2 = OC/4
OC = 2
Pythagorean theorem: c^2 = a^2 + b^2
4^2 = 2^2 + b^2
b^2 = 16 - 4 = 12
b = √12 = 3.46 or BC = AC = 3.46
=> AB = 3.46 x 2 = 6.92
A=1/2hb
Area of triangle ABO = 1/2(OC)(AB) = 1/2(2)(6.92) = 6.92
Area of the shaded segment = area of circle - (1/2 area of circle + area of 60o + area of triangle)
=> π4^2 - (8π + 8/3π + 6.92)
=> 16π - 8π - 8/3π - 6.92
=> 16/3π - 6.92
=> 16/3(3.14) - 6.92 = 9.83