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I have an equation to calculate $A$ from $r$ and $h$: $$ A= \arccos\left(\frac{r-h}{r}\right)r^2 - (r-h)\sqrt{r^2-(r-h)^2} $$

But now $A$ and $r$ are known, and I want to solve for $h$ instead. So far, I solve this equation numerically, but I was wondering whether it can be solved analytically. All three $A$, $r$ and $h$ are positive. I have tried to solve it using Wolfram Alpha, but I only get a timeout. Any better ideas?

matth
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    Well, that's essentially Kepler's equation (https://en.wikipedia.org/wiki/Kepler%27s_equation#Radial_Kepler_equation).

    " I am sufficiently satisfied that it [Kepler's equation] cannot be solved a priori, on account of the different nature of the arc and the sine. But if I am mistaken, and any one shall point out the way to me, he will be in my eyes the great Apollonius."

    — Johannes Kepler

     –  Jan 03 '21 at 15:37
  • Thanks, I came across this equation when reviewing code for calculating areas, the Wikipedia link has some helpful links, e.g. this one: https://www.winemantech.com/blog/the-numerical-analysis-of-finding-the-height-of-a-circular-segment – matth Jan 04 '21 at 09:43

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