= Area of a regular octagon
{wiki}
A <regular polygon> with eight sides, each of <length> $s>0$, has <area>
$$
A=2(1+\sqrt2)s^2.
$$
Join its center to all eight <vertices of a polygon>. The resulting congruent <isosceles triangles> each have base $s$ and central <angle> $\pi/4$. Bisecting one gives a <right triangle> with opposite side $s/2$ and adjacent side $h$, the perpendicular center-to-side <distance>. Hence
$$
h=\frac{s}{2\tan(\pi/8)}=\frac{1+\sqrt2}{2}s.
$$
Adding the eight <triangle> <areas> gives $A=8(sh/2)=4sh$, proving the formula. For side <length> one, the <area> is $2+2\sqrt2$.
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