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The goal is to design a container (partial sphere) of given volume which attached to a plane via a port of a given radius. I believe this can be done as follows but the calculation is causing me problems:

A circle of unknown radius is split by a chord of known length L. The larger circular segment has known area A. Given this information, is it possible to calculate the radius of the circle? To clarify, it is also given that L is not a diameter of the circle.

I believe whatever approach is used for the above would also be extensible to a spherical cap.

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Image from wikipedia

With the notations in this figure (borrowed from https://en.wikipedia.org/wiki/Circular_segment), $c$ is your chord length $L$, $$\theta=2\arcsin(\frac{c}{2R}),$$ the green area is $\frac{R^2}{2}(\theta-\sin(\theta))$. The larger circular segment area (whole area of the disk minus the green area) is given by $$A=R^2(\pi-\theta/2+\sin(\theta)/2).$$ If you replace $\theta$ from the first equation, you obtain a transcendental equation in $R$. It is possible to solve this numerically.

Similar equations for a spherical cap can be found at https://en.wikipedia.org/wiki/Spherical_cap

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