I am making a simple 3D printed mechanism that consists of a tab and housing. Both the tab and housi

I am making a simple 3D printed mechanism that consists of a tab and housing. Both the tab and housing have nubs that interface with each other, preventing the tab from exiting the housing. The nubs are circular arcs with height 0.5mm and length 3.00mm. Fusion 360 calculated this for me to 4 decimal places, but I want to know the formula I can use, given the height and nub length, in order to find the exact value for the distance. This is necessary because I need the formula in order to use the output value for other aspects of the design.
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enfujahl
Step 1
For an arc height of h and width w, the radius of the arc satisfies
${r}^{2}=\left(r-h{\right)}^{2}+\left(w/2{\right)}^{2},$
Hence
$r=\frac{4{h}^{2}+{w}^{2}}{8h}.$
Consequently, the horizontal separation d between the two marked extrema of the arcs satisfies
$\left(2r-h{\right)}^{2}+{d}^{2}=\left(2r{\right)}^{2},$
or
$d=\sqrt{4hr-{h}^{2}}=\sqrt{{h}^{2}+\frac{{w}^{2}}{2}}.$
Hence the total width of the configuration as a function of h and w is
$d+w=w+\sqrt{{h}^{2}+\frac{{w}^{2}}{2}}.$
For the example shown, $h=0.5$ and $w=3$ yields $d+w=3+\frac{\sqrt{19}}{2}\approx 5.17944947$ , which agrees with the yellow highlighted value.

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