equable regular polygon formula

for any regular polygon P with side length s and side count n, the side length s necessary such that its perimeter P_P and area A_P are numerically equivalent may be found using the formula:

$s = 4\tan{\frac{\pi}{n}}\;,\;\mathrm{when}\;\{n\in\mathbb{N}\mid3\leq n\}$

which is derived from the following:

$\mathbb{{D}}:\mathbb{R}\setminus\mathbb{Z}^-$
$F=\{P\}$
$\forall{P}\in{F}\;:\;\exists{a=\frac{s\cot\frac{\pi}{n}}{2}}$
$\forall{P}\in{F}\;:\;\exists{A_P=\frac{ans}{2}}$
${A_P=\frac{ns^2\frac{\cot\frac{\pi}{n}}{2}}{2}$
$A_P=\frac{ns^2\cot\frac{\pi}{n}}{4}$
$\forall{P}\in{F}\;:\;\exists{P_P=ns}$
$\exists{P}\;:\;A_P=P_P$
$H=\{P\mid{A_P=P_P}\}$
$P\in{H}\iff{A_P=P_P}$
$\therefore\;P\in{H}\Longrightarrow{\frac{ns^2\cot\frac{\pi}{n}}{4}=ns}$
$\frac{s^2\cot\frac{\pi}{n}}{4}=s$
$\frac{\cot\frac{\pi}{n}}{4}=\frac{s}{s^2}$
$\frac{\cot\frac{\pi}{n}}{4}=\frac{1}{s}$
$\frac{4}{\cot\frac{\pi}{n}}=s$
$4\tan\frac{\pi}{n}=s$
$\Box$