Find the area of the shaded regions below. give your answer as a completely simplified exact value in terms of π (no approximations). ( can you explain how you did it)

Hello from MrBillDoesMath!

  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!

Thank you,

MrB