# How to calculate the height of a circular segment based on the area.

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?

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 $h$.
• @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 – Samrat Mukhopadhyay Dec 3 '14 at 10:09
• @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 – Samrat Mukhopadhyay Dec 3 '14 at 17:08