Can't find exact value of $x = \frac{\pi}{\cot^{-1}(\frac{4\pi}{x})}$

Question

I can't find the exact value of x for the equation $x = \frac{\pi}{\cot^{-1}(\frac{4\pi}{x})}$. I tried typing it into wolfram alpha and it only gave me a decimal approximation, approximately equal to 6.543697264. If you click this link https://www.desmos.com/calculator/rcf54bxjsz, then when t equals the number in the photo then the yellow line exactly matches the dotted black line, which is the formula for finding a circles radius if you only know its area. image here

Answer 1

What do you mean by "exact solution"?

A lot of numbers are much easier to express as the equations they solve than a closed form of given functions or algebraic expressions.

Maybe there does not exist a closed form expression. In fact it is often so much easier to express numbers as equations it can often be a means of quite efficient data compression.

For example $x^2+y^2 = 1$ is fulfilled by all numbers on the unit circle. How would you capture that solution set in any easier or more compact way?

Sometimes the right thing to do is not to find a solution set for an equation but rather the best equation which matches the solution set.