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:
enter image description here
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):

enter image description here

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[8], t], {t, -4, 4}, 
 PlotStyle -> {Dashed, Hue[#]}, PlotRange -> {{-1.025, 1.025}, {-.025, 
               2 Sqrt[2] + 0.025}}] & /@ (Range[m]/m), 
 ParametricPlot[G[Sqrt[8], t], {t, -1, 1}, PlotStyle -> {Red, Thick}], 
 Graphics[{Black, Disk[{0, 1}, .025], Opacity[0.1], Rectangle[{-1, 0}, {1, 2}],
           Opacity[1], Line[{{c, 0}, G[Sqrt[8], c]}], Disk[{c, 0}, .025],
           {Hue[#], Disk[G[# Sqrt[8], c], .025]} & /@ (Range[m]/m)}],
 Axes -> False]
Manipulate[P[c, m], {c, -1, 1}, {m, 1, 20, 1}]

Source : Link , Question Author : Pontus Magnusson , Answer Author : heropup

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