Bounding the Circle

The area of a circumscribed regular $n$ -gon consists of $n$ triangles with width $2 \tan (180^\circ/n)$ and height 1: $A_{circsc} = n\cdot \tan \left(\frac{180^\circ}{n}\right).$ The inscribed regular $n$ -gon can likewise be split into $n$ triangles with width $2\sin (180^\circ/n)$ and height $\cos (180^\circ/n)$ , so that $A_{insc} = n\cdot \sin \left(\frac{180^\circ}{n}\right) \cos \left(\frac{180^\circ}{n}\right).$

Substituting $n = 16$ we find the values $A_{circsc} \approx 3.182598; \ \ A_{insc} \approx 3.061467.$ Subtraction gives a range of $\boxed{0.121130}$ .

Alternatively, the range is equal to $n\cdot \tan \left(\frac{180^\circ}{n}\right) \cdot \sin^2 \left(\frac{180^\circ}{n}\right).$

We generalise it for a regular n-sided polygon.

$Range = A_{circsc} - A_{insc}$

$\displaystyle \Rightarrow Range = nr^{2} \displaystyle \tan(\frac{\pi}{n}) - \frac{nr^{2}\displaystyle \sin(\frac{2\pi}{n})}{2}$

Substituting n=16 and r=1, we get :

$\displaystyle \Rightarrow Range = 16\times \Big( \tan(\frac{\pi}{16}) - \frac{\sin(\frac{\pi}{8})}{2} \Big)$

$\displaystyle \Rightarrow Range = \boxed{0.121130419}$

$A_{insc}=\frac n 2*R_{insc}^2Sin\dfrac{360}n.\\ A_{circsc}=\frac n 2*R_{circsc}^2Sin\dfrac{360}n , ~~but ~1=R_{incsc}=R_{circsc}Cos\dfrac{180}n .\\ \implies~ A_{circsc}=\frac n 2*\left (\dfrac{R_{insc}}{Cos\dfrac{180}n}\right )^2*Sin\dfrac{360}n.\\ \therefore ~ Range=A_{circsc} - A_{insc}=\frac n 2*R_{insc}^2*Sin\dfrac{360}n \left \{ \left (\dfrac 1 {Cos\dfrac{180}n}\right )^2 - 1 \right \}\\ Range=\frac 1 2*16*1^2*Sin\dfrac{360}{16} \left \{ \left (\dfrac1{Cos\dfrac{180}{16}}\right )^2 - 1 \right \}=.1211304192$