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Answer: 8 (Pi - sqrt(3))
Discussion:
The area of the shaded region is that of the semicircle minus the area of the triangle..
Area of semicircle = 1/2 * Pi * R^2
Where R^2 is the square of the radius of the circle. In our case, R ( = OC)
= 4 so the semicircle area is
(1/2) * Pi * (4^2) = (1/2) * Pi * 16 = 8 Pi
Area of triangle.
First of all, angle ACB is a right angle ( i.e. 90 degrees).
* This is the Theorem of Thales from elementary Plane Geometry. *
so by Pythagoras
AC^2 + BC^2 = AB^2
But CB = 4 (given) and AB = 4*2 = 8 ( the diameter is twice the radius).
Substituting these in Pythagoras gives
AC^2 + 4^2 = 8^2 or
AC^2 = 8^2 - 4^2- = 64 - 16 = 48
Hence AC = sqrt(48) = sqrt (16*3) = 4 * sqrt(3)
We are almost done! The area of the triangle is given by
(1/2) b * h = (1/2) BC * AC = (1/2) 4 * (4 * sqrt(3)) = 8 sqrt(3)
We conclude the area area of the shaded part is
8 PI - 8 sqrt(3) = 8 (Pi - sqrt(3))
Note that sqrt(3) is approx 1.7 so (PI - sqrt(3)) is a positive number, as it better well be!
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