How to find the area of the red-colored enclosure?

Question

I am helping someone prepare for an exam. And there came up this problem.

• What is given is:

  1. The angle α is 90°.
  2. The two sides of the triangle have a length a, i.e. it is an isosceles triangle.

• What we can infer (or at least what I tried to infer):

  1. The semi-circle divides the side of the triangle equally into two parts of length a/2.
  2. The perpendicular from the point of the semi-circle to the base of the triangle would be also then a/2 long.
  3. Thus the radius of the semi-circle is a/2 and hence the area of the semi-circle is:

π(a/2)²

4. Now since the 2 sides of the triangle are of the same length a and the angle α is a right angle, the lenght of the base of the triangle can be found as:

$a*\sqrt{2}$

5. The area of the right-angled triangle can be calculated as:

1/2 (base) x (height)

which is,

1/2 (a*a)

So far so good. I have the area of the semi-circle and the area of the triangle.

Now how to find out the area of the red-shaded portion? I don't have any further hints or anything in the problem statement.