Distance Across Two Tangential Arcs

Question

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. This picture illustrates what it looks like when the nubs come in contact with each other: The nubs are circular arcs with height 0.5mm and length 3.00mm as pictured. The two arcs are tangential to one another, touching at only one point. The distance I want is highlighted in yellow. 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.

Answer 1

For an arc height of $h$ and width $w$, the radius of the arc satisfies $$r^2 = (r-h)^2 + (w/2)^2,$$ hence $$r = \frac{4h^2 + w^2}{8h}.$$ Consequently, the horizontal separation $d$ between the two marked extrema of the arcs satisfies $$(2r-h)^2 + d^2 = (2r)^2,$$ 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.