area of a lune

[dolor]

A semicircle of diameter 1 sits atop a semicircle of diameter 2, as shown. The unshaded area inside the smaller semicircle and outside the larger semicircle is called a lune. Determine the area of the lune.

[opus]

So you know the size of the semicircle 1.
All you need is the size of the circle-segment of 2 that is within circle 1.
You need to know to diameter of Circle 1 and 2 and the relative position of both.
Having that you can calculate the position of the two crossing points of circle 1 and 2, noting the angle from the centre of circle 1.

Using these angles you can calculate the circle segment of circle 2 that is bound by the two crossing points.
Using the positions of the crossing points and the centre of circle 2, you can calculate the triangle formed by those points.
Subtract the area of the triangle from the circle segment and you have the Shaded part of circle 1.
Now subtract from the size of Circle 2 and you have the answer.