Dido problem for a circle [closed]

Could anyone find the solution for this generalization for Dido problem: given a big circle island with radius $R$ and line with fixed length $l << R$ we want to find the maximal area bounded by this line and the border of island?

So I've been thinking on this problem for couple of days and haven't solution yet. I think the solution is some arc of circle, but I can't find the maximum of the resulting formula for area (and also Wolfram Mathematica can't, or maybe I do something wrong in it).

Maybe it will be helpful to solve Dido problem for island with angle shape.

Or maybe some Euler-Lagrange equations for this problem are known?

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it is the arc of circle of given length, that meets orthogonally the border. –  Pietro Majer Apr 10 at 9:15
Yes, I think, it's true. But why? –  ptashek Apr 10 at 10:10
This question appears to be off-topic because it would be appropriate on math.stackexchange, but is not about research mathematics. –  Theo Johnson-Freyd Sep 1 at 1:50