Astronomers really want to be able to know how big, physically, stars are. Our understanding of how stars behave (like how they shine, how hot they are, how bright they are, and even (especially) how big planets are that are orbiting them) depends a lot on how big they are.
Now, you can figure this out if you know how far away they are and how big they appear to be. A little high school trig solves that problem easily. But how do you measure their apparent size, how big they seem to be in the sky?