How do I spawn objects within only a circle radius?

So I'd like for my game to spawn objects only within a circle..

The only way I've found that I can do this is if I spawn all my objects within a square and have an inverted circle sprite that destroys all the overlapping objects on collision.. however it seems quite ineffective and doesn't happen immediately.

The above image is currently what I'm left with. I am hoping there is a better solution to this, I've searched the forums however a lot of the old examples are broken links/ similar solutions to what I have now.

The trig formula is x+cos(angle)*distance, y+sin(angle)*distance.

dropbox.com/s/fde1lxyoup6a8z7/spawncirc.c3p

Exactly what I was looking for. Thank you so much for including both a written answer and a demo project.