Given an area of a circular segment, how can one find the height of the circular segment?

In the image below, assume the area of the green segment is known. How can one find the value of h?

Circular Segment

I have also seen this problem described as the Quarter Tank Problem.

Is there a way to solve this problem without recursive approximation?


The area of the green portion will be $\displaystyle A=\frac{1}{2}\theta R^2-\frac{1}{2}R^2\sin \theta$ Also, you have $$d=R \cos \left(\frac{\theta}{2}\right)\\ h=R-d=R\left(1-\cos \left(\frac{\theta}{2}\right)\right)$$ So given $A,R$ you have to solve the transcendental equation $$ A=\frac{1}{2}\theta R^2-\frac{1}{2}R^2\sin \theta$$ to get $\theta$. Then you can compute $h$.

  • $\begingroup$ Sure @DonAntonio . $\endgroup$ – RicardoCruz Jul 23 '13 at 18:54
  • $\begingroup$ Is there a closed form solution to the transcendental equation? If so, could you provide it or provide a resource that will allow me to solve it? $\endgroup$ – Tarik Dec 2 '14 at 11:49
  • $\begingroup$ @Tarik, there is no closed form solution of this transcendental equation. You can solve it numerically. Since $0<\theta<\pi$, this function is a convex function and hence you are guaranteed to find a unique solution using Newton's method $\endgroup$ – Samrat Mukhopadhyay Dec 3 '14 at 10:09
  • $\begingroup$ Thanks @SamratMukhopadhyay for your answer. Can we assert that a given type of equation does not have a closed form solution? If so, how? Can we assert that there exists a closed form solution to an equation although unknown at this time? Or shall we assume that some classes of equations may have a closed form solution but we do not know for certain. Could we solve this particular equation in terms of one of the known mathematical special functions? If not, would it make sense to create a new special function in terms of which a whole set of transcendental equations could be solved? $\endgroup$ – Tarik Dec 3 '14 at 11:16
  • $\begingroup$ @Tarik it is true that there can be equations which may seem to be not having closed form solution at a first glance but may be solved by specialized function. Actually we need to see if there are functions, useful for the equation, which are well studied. For an example the equation $z=e^z$ is a transcendental equation but it has a "closed" form solution in terms of Lambert's W function $\endgroup$ – Samrat Mukhopadhyay Dec 3 '14 at 17:08

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