Question

Hey ^-^ can someone please help me with this problem:

Find the area of the shaded regions below. Give your answer as a completely simplified exact value in terms of π (no approximations).

Answer 1

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!