5. Let $[x]$ denote the greatest integer not exceeding the real number $x$. Then, in the plane, the area of the figure formed by all points satisfying $[x]^{2}+[y]^{2}=50$ is

保留源文本的换行和格式，直接输出翻译结果。
5.12.

First, consider the first quadrant.
From $50=1^{2}+7^{2}=7^{2}+1^{2}=5^{2}+5^{2}$, we get $([x],[y])=(1,7),(7,1),(5,5)$.
And from $[x]=1,[y]=7$, we get the unit square $1 \leqslant x<2,7 \leqslant y<8$, whose area is 1.

Similarly, from $([x],[y])=(7,1)$ and $(5,5)$, we also get one unit square each, and these three squares do not overlap.

If we consider all four quadrants, we get a total of 12 non-overlapping unit squares, with a total area of 12.