Sqrt is actually one of the faster operations, though obviously avoiding the need for any operation is clearly preferable :)
Here's a much simpler way to get the same distribution (in JavaScript syntax):
var radius1 = Math.max(Math.random(), Math.random());
var radius2 = Math.sqrt(Math.random());
Whether taking the max of two variables is cheaper than the square-root of one depends on implementation details, I guess?
You can also re-use the random variable that the max 'tossed' away by scaling it.
If you can re-jig the rest of your algorithm to consume the square of the radius (instead of the radius itself), you don't need to take a square-root here.