Once you have constructed everything accurately, the Pathfinder part of it is easy:
1. Draw as many circles as you need. Here I used compound paths and the pink is just to show the “holes”.
2. Lock the big circle. Use Pathfinder Divide on the small circles. Select all unpainted objects and delete them.
3. Delete all the parts you don’t need. Note that there are several little pieces near the intersections of the circles that you will need to remove carefully.
4. Use Pathfinder Unite on the remaining pieces. unlock the big circle and Group the whole thing.