Toolkit 86

Regular Polygons

For an $n$ -sided polygon,

\text{Sum of interior angles}=180^\circ(n-2)

For a regular hexagon with side length $a$ :

\angle A=\frac{180^\circ(6-2)}{6}=120^\circ

[ABCDEF]=6\left(\frac{a^2\sqrt{3}}{4}\right)=\frac{3a^2\sqrt{3}}{2}

AC=a\sqrt{3}

AD=2a

For a regular octagon with side length $a$ :

\angle A=\frac{180^\circ(8-2)}{8}=135^\circ

[ABCDEFGH]=(a+a\sqrt{2})^2-4\left(\frac{a^2/2}{2}\right)

=a^2+2a^2+2a^2\sqrt{2}-a^2=2a^2(1+\sqrt{2})

AC=a\sqrt{2+\sqrt{2}}

AC^2=a^2+a^2-2a^2\cos 135^\circ=2a^2+2a^2\frac{\sqrt{2}}{2}

=2a^2+a^2\sqrt{2}=a^2(2+\sqrt{2})

AD=a\sqrt{2}+a=a(\sqrt{2}+1)

AE=a\sqrt{4+2\sqrt{2}}

AE^2=a^2+AD^2=a^2+\left(a(\sqrt{2}+1)\right)^2

=a^2+a^2(3+2\sqrt{2})=a^2(4+2\sqrt{2})

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