Question

In the diagram, AYB is a semicircle with AB as diameter and OAXB is a sector of a circle with centre  O and radius r. Angle AOB = 2 radians. Find an expression, in terms of r and , for the  area of the shaded region.

Solution

It is evident from the diagram that;

First we calculate area of semi-circle AYB.

Expression for area of semi-circle with radius  is;

Therefore, we need to find radius of semicircle AYB.

We are given that AB is the diameter of the circle AYB. Hence;

It is evident from the diagram that  is an isosceles triangle with congruent sides   and base . The vertex angle of this isosceles triangle is . 

Now consider the diagram below to find the radius of the circle AYB.

In an isosceles triangle, the bisector of the vertex angle (the angle which is opposite to a side that might not be congruent to another side) is also perpendicular bisector of the base.

Therefore,  is an angle bisector of the  and is also perpendicular bisector of base . Hence;

Now consider the two triangles AOS and BOS.
It is evident that  and  and .

Law of Sines;

Applying law of sines to ;

Now we can find area of semi-circle AYB.

Next we calculate area of segment AXB.

It is evident from the diagram that;

Let’s first find area of circular sector AXB and then area of triangle AOB.

Expression for area of a circular sector with radius  and angle  rad is;

It is evident that sector AXB is sector of the circle with center O which has radius  and angle of circular sector is . Therefore;

Lastly, we need area of triangle AOB.

Expression for the area of a triangle for which two sides (a and b) and the included angle (C ) is  given;

For the given case, in ;

Finally;