Estimate for the co-volume of discs centered at lattice points in the plane?

Question

Suppose I have a unimodular lattice $\Lambda = A \mathbb{Z^2}$ ($A\in SL(2,\mathbb{R})$) in the plane. I place a disc of fixed radius, $r$, around each point of $\Lambda$, so that I have a union of (possibly overlapping) discs $$\bigcup_{v\in \Lambda} \mathbb{D}^2(v, r) := \mathfrak{D}$$ and I want to estimate the covolume of $\mathfrak{D}$, i.e. the measure of $\mathfrak{D}\cap P$, where P is any fundamental parallelogram. Are there any known estimates for this covolume in terms of $A$ and $r$? I'm quite out of my depth, so any vague suggestions are welcome. Thanks.