AFAIK there is no easy way (if you rule out the normal trim in the sketch, which can get confusing when you have too much circles), however i found a workaround that requires more work.
check part attached.
1- Create sketch and draw in it only one circle
2- Extrude the circle
3-Right click on sketch and click share sketch
4-Edit your sketch and add one other circle, then hit finish sketch
5-Extrude your second circle, this time using "intersect" instead of "Join" (Make sure your extrusion is in the right direction, otherwise you'll get nothing).
6- Re-edit your sketch and create a third circle, and extrude "intersect".
And so on until you get your solid made of intersections, then you can project its face to any plane.
If you find it hard let me know, i'll try to create some video for you tomorrow.
Best Regards,
