What is an appropriate cut radius for a semi-circle bench for a 36" round table?
Table over hang is anywhere from 0-5 inches judging from these pictures' measurements and eyeballing what I've seen online. Meaning the seat is the same radius or smaller. If you call it at a 4" over hang per side then you need a 28" radius semi-circle cut in the bench. I slouch, so 2-4" is good for me. People with good posture probably expect 4-6". 

Experiment with a chair and table to find your preference. Go to a restaurant that has booths, take some measurements and scale them to your table. I wouldn't make it any smaller though, unless you only expect your kids to sit there. I've never seen a corner booth smaller then say 4'. I'd better really like the dude I'm sharing my 28" circle of knee space with. 

Your question was asked here: gardenweb.com, specifically suggesting to pay  attention to the seat back distance from the table, to accommodate normal(?) sized humans. The differential of the radii is open to interpretation, the space between the table and the back of the seat isn't, generally to be not less than 18". (example 1 has it at 21")


http://www.centralrestaurant.com/bg-spaceplanning.html




https://m1.behance.net/rendition/modules/6133285/hd/ad04c5d37a255529d55320cc9d96b257.jpg





http://www.pinterest.com/pin/161074124148336216/

It doesn't seam like you'll be having a fixed table, and if the booth is built wrong, guests will be continually deciding 'where it goes'.