In the figure below, a quarter-circle is inscribed in the big circle of radius 10. The top boundary curve of the shaded region is made by a circle with a center at the top of the shown circle. What is the area of the shaded region?

The shaded region is:

If we want to find the area of, we should find the area of first.

.

We should use the following figures to find.

.

To find, we should find R first. Since R is the base and height of the triangle,

, and so.

The central angle of S_{4} = 90 because it is
the angle measurement of 2 45^{o} angles. Thus, since R^{2} =
200,

.

Thus,

And

.