Question

Find the area of the regular 20-gon with radius 6 mm.

Answer 1

now, by radius of a polygon, we're referring to the distance from its center to a corner where two sides meet, or namely the radius of the circle that surrounds it or namely the circumcircle.

`\underset{ \textit{angle in degrees} }{\textit{area of a regular polygon}}\\\\ A=\cfrac{nR^2}{2}\cdot \sin((360)/(n)) ~~ \begin{cases} n=sides\\ R=\stackrel{\textit{radius of}}{circumcircle}\\[-0.5em] \hrulefill\\ n=20\\ R=6 \end{cases}\implies A=\cfrac{(20)(6)^2}{2}\cdot \sin((360)/(20)) \\\\\\ A=360\sin(18^o)\implies A\approx 111.25~mm^2`

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