Isolating $n$ in $A = \frac {nL^2}{4\tan(180^\circ/n)}$

Question

$$A = \frac {nL^2}{4\tan(180^\circ/n)}$$

I want to isolate $n$, but why I try, I can't get it to one side.

Thank you for your help!

Answer 1

There is not explicit solution to the equation and numerical methods would be needed.

Let $x=\frac \pi n$ and $k=\frac {L^2 \pi}{4A}$ to make it $$k=x \tan(x)$$ which has an infinite number of solutions. Let us consider that we just look for the first positive solution.

Assuming $x$ to be "small", for an approximation, we can build at $x=0$ a Padé approximant of $x \tan(x)$ to get $$k\simeq \frac{x^2-\frac{2 }{21}x^4}{1-\frac{3 }{7}x^2+\frac{1}{105}x^4}$$ which let us with a quadratic equation in $x^2$.

Trying for $k=1$, this would give $x=\sqrt{\frac{1}{11} \left(75-\sqrt{4470}\right)}\approx 0.860335$ while the exact solution would be $0.860334$

Trying for $k=10$, this would give $x=\frac{1}{2} \sqrt{\frac{1}{2} \left(111-\sqrt{8961}\right)}\approx 1.42905$ while the exact solution would be $1.42887$.

For better approximations, have a look at this question of mine which addresses exactly the problme of this equation.