I'm not a mathematician, but perhaps the following explanation may help...

This formula

r = sqrt( x^{2} + y^{2} )

computes the (Euclidian) distance of the 2D point (x,y) from the origin (0,0). The distance alone isn't sufficient to define a concrete point; i.e. it lacks the direction along which the distance could be drawn. That means that the single formula describes not a single but (infinitely) many points. As a sentence:

"Each point (x,y) in space for which the distance is equal to a given distance r belongs to a circle of radius r concentric to the origin; each other point that does not match the equation does not belong to that circle."

Notice that if you would have another criterion, e.g. the said direction, you have another infinite set, in the example "all points of space that lie on a half line emanating from the origin...". Intersection both sets yields in a single point again, identical to (x,y) but described within another co-ordinate system (distance and direction, in this case).

However, you are interested in the set of points, not a single point.

Similarly, a disc can be expressed as: sqrt( x^{2} + y^{2} ) <= r

And a ring can be expressed as: r_{i} <= sqrt( x^{2} + y^{2} ) <= r_{a}

But I couldn't get my head to wrap around how to increase the radius without touching the X and Y values.

Choosing another r means that, according to the above explanation, from all possible (x,y) another subset is chosen. Understand r as independent variable, and (x,y) as dependent variable.