Producing circles with $r^2=x^2+y^2$ $r^2=x^2+y^2\ +\pi $ produces graph with two circles :
$r = 2$

This graph was produced using Desmos, is this correct or a quirk of the graph software ?
Ive noticed $\forall n \epsilon N [r^2=x^2+y^2+n\pi]$ appears to produce same circle, so $r^2=x^2+y^2\ +3\pi $ produces same circle as $r^2=x^2+y^2\ +4\pi $  :

Why is this the case, should each circle not be differing sizes ?

Your first picture is correct if you are graphing
\begin{align*}
2^2 &= x^2 + y^2\\
- \pi + 2^2 &= x^2 + y^2
\end{align*}
for red and yellow respectively. If we have the equation
$$
r^2 = x^2 + y^2 + c
$$
where $c$ is a constant, that is the same as
$$
r^2 - c = x^2 + y^2
$$
and so our new radius is $r' = \sqrt{r^2 - c}$, if $c > r^2$, the circle has no points as squares are positive. As $3\pi$ and $4\pi$ are larger than $2^2$, there are no solutions for
\begin{align*}
-3\pi + 2^2 = x^2 + y^2\\
-4\pi + 2^2 = x^2 + y^2.
\end{align*}
Desmos will therefore not draw anything.
