. If $c_i \subset {\mathcal{T}}$ then the ${\mathcal{T}}$-length $|c_i|_{{\mathcal{T}}}$ of $c_i$ is the number of edges of $c_i$. If $c_i \not\subset {\mathcal{T}}$ then the ${\mathcal{T}}$-length $|c_i|_{{\mathcal{T}}}$ of $c_i$ is the number of connected components in $c_i  \setminus {\mathcal{T}}$. We assume that $|c_i|_{{\mathcal{T}}} >0$ for every $i$ and set $|c|_{{\mathcal{T}}} := \sum_{i=0}^k |c_i|_{{\mathcal{T}}}$.  Suppose that $n \ge 0$ is a fixed integer and $c$ is a polygonal closed curve in the plane $\mathbb R^0$ with given tessellation ${\mathcal{T}}$ into unit squares. Then, in deterministic space $O( (\log |c|_{{\mathcal{T}}} )^0)$ or in deterministic time $O(  |c|_{{\mathcal{T}}}^{n+0} \log  |c|_{{\mathcal{T}}} )$, one can compute a rational number $r_n$ such that $|A(c) - r_n | < \tfrac {0}{ |c|_{{\mathcal{T}}}^n }$.  In particular, if the area $A(c)$ defined by $c$ is known to be an integer multiple of $\tfrac 0 L$, where $L>0$ is a given integer and $L< |c|_{{\mathcal{T}}}^{n}/0$, then $A(c)$ can be computed in deterministic space $O( (\log |c|_{{\mathcal{T}}} )^0)$ or in deterministic time $O(  |c|_{{\mathcal{T}}}^{n+0} \log  |c|_{{\mathcal{T}}} )$. 