Harristotelian Logic

Here's something where I was working on one thing, did something wrong, so I came up with this conjecture, and later realized that it didn't follow from what I was working on originally so I dropped it, but find myself curious enough to pose it as a question.

Given a circle

a^2 + b^2 = r^2

with an integer radius what is the frequency with which you will have non-zero integer solutions?

Here's an example to understand the question as consider the first five cases:

3^2 + 4^2 = 5^2

6^2 + 8^2 = 10^2

5^2 + 12^2 = 13^2

9^2 + 12^2 = 15^2

and

8^2 + 15^2 = 17^2

So you have 5 of the first 17 naturals, and originally I had thought the ratio would be about 1/3, but some posters replying to an original post of mine on this subject claimed otherwise.

What is the actual ratio?

It's the kind of question that I'd think would have been asked before, so alternatively, can someone provide a link to an answer?

Oh yeah, secondary would be what is the ratio for primitive solutions i.e. taking away the multiples of another solution like 6^2 + 8^2 = 10^2?