I want to enter a quantity for plane angle that I will convert to radians and degrees

Why don't you just use radians or degrees?

As for searching for supported quantities, I'm not aware of an easy way to do that. But you already know that radians and degrees are supported.

This is a long shot, but maybe you'd be interested in FormulaLookup. You can specify a RequiredPhysicalQuantities option.

I think I must be missing something, because it sounds like you already know how to solve your problem. Maybe you can restate your question. Are you asking how to interpret "semicircular arc with a radius of 220 m"? Or how to turn that into an arclength? Maybe you can narrow down the text of the problem to specifically what you're asking about.