Fill the region bounded by two path in TikZ

Question

How to fill the region (with gray!50) bounded by two path in TikZ? I have tried many solutions, but they all don't work.
begin{tikzpicture}[scale=1.7]
draw[semithick] (0,0) -- (2,0) -- (2,2) -- (0,2) -- cycle;
draw[name path=f, semithick] (1,1) circle (1);
draw[name path=g, semithick] (0,0) arc (270:360:2);
end{tikzpicture}

Namely, I want something like this:

user230294 · Accepted Answer

documentclass[tikz,border=3mm]{standalone}
usetikzlibrary{intersections}% <- not needed for the below code
begin{document}
begin{tikzpicture}[scale=1.7]
begin{scope}
 clip  (1,1) circle [radius=1];
fill[gray]  (0,0) arc[start angle=270,end angle=360,radius=2] |- cycle;
end{scope}
draw[semithick] (0,0) rectangle (2,2);
draw[name path=f, semithick] (1,1) circle [radius=1];
draw[name path=g, semithick] (0,0) arc[start angle=270,end angle=360,radius=2];
end{tikzpicture}
end{document}

hpekristiansen · Answer

clip makes everything following only drawn inside the clip path.
documentclass[tikz, border=1cm]{standalone}
begin{document}
begin{tikzpicture}
draw (0,0) rectangle (2,2);
draw (0,0) arc (270:360:2);
draw[clip] (1,1) circle (1);
fill[gray!50] (0,0) arc (270:360:2) -- (2,0) -- cycle;
end{tikzpicture}
end{document}

Alain Remillard · Answer

An other way to do it, without the clip command, is to fill the region first, then draw the lines.
documentclass[tikz, border=1cm]{standalone}
usetikzlibrary{intersections}
begin{document}
begin{tikzpicture}[scale=1.7]
fill[gray!50] (1,1) circle (1);    % gray circle
fill[white] (0,0) arc (270:360:2) -| cycle;    % white sector over
draw[semithick] (0,0) -- (2,0) -- (2,2) -- (0,2) -- cycle;
draw[name path=f, semithick] (1,1) circle (1);
draw[name path=g, semithick] (0,0) arc (270:360:2);
end{tikzpicture}
end{document}

This gives the following

Fill the region bounded by two path in TikZ

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