# Area covered by a constant length segment rotating around the center of a square.

This is an idea I have had in my head for years and years and I would like to know the answer, and also I would like to know if it’s somehow relevant to anything or useless.
I describe my thoughts with the following image: What would the area of the “red almost half circle” on top of the third square be, assuming you rotate the hypotenuse of a square around it’s center limiting its movement so it cannot pass through the bottom of the square.
My guess would be:

And also, does this have any meaning? Have I been wandering around thinking about complete nonsense for so many years?

I found this problem interesting enough to make a little animation along the line of @Blue’s diagram (but I didn’t want to edit their answer without permission): Mathematica syntax for those who are interested:

G[d_, t_] := {t - (d t)/Sqrt[1 + t^2], d /Sqrt[1 + t^2]}
P[c_, m_] := Show[ParametricPlot[G[# Sqrt, t], {t, -4, 4},
PlotStyle -> {Dashed, Hue[#]}, PlotRange -> {{-1.025, 1.025}, {-.025,
2 Sqrt + 0.025}}] & /@ (Range[m]/m),
ParametricPlot[G[Sqrt, t], {t, -1, 1}, PlotStyle -> {Red, Thick}],
Graphics[{Black, Disk[{0, 1}, .025], Opacity[0.1], Rectangle[{-1, 0}, {1, 2}],
Opacity, Line[{{c, 0}, G[Sqrt, c]}], Disk[{c, 0}, .025],
{Hue[#], Disk[G[# Sqrt, c], .025]} & /@ (Range[m]/m)}],
Axes -> False]
Manipulate[P[c, m], {c, -1, 1}, {m, 1, 20, 1}]