Regular Polygons - Decomposition into Triangles

Regular polygons mean all sides $S$ are equal and so are all angles. They have a circumcircle with radius $R$ and an incircle with radius $r.$

A regular polygon can be analyzed easily if we think of it as having been built from isosceles triangles with the unequal side $AB=S,$ the equal sides $OA=OB= R,$ and the unequal angle $\angle AOB=2\phi$ at the center $O$ of the circumcircle. Drop the perpendicular $OM=r$ from $O$ to the base $AB,$ where $OM$ is the inradius.

With $n$-sided polygon, we have the following relations among these important quantities:

• $\phi=\dfrac \pi n$
• $R=\dfrac{\frac S 2 }{\sin\phi} = \dfrac{ S }{2\times \sin\phi}\quad$ or $\quad S=2\times R\times \sin\phi$
• Area of the $\triangle =\dfrac 1 2 \times S\times p=\dfrac 1 2 \times S\times \dfrac S 2\times \cot \phi.$

So, remember $$\phi=\dfrac \pi n,\quad R= \dfrac S {2\times \sin\phi},\quad S= 2\times R\times \sin\phi ,\quad \text{Area}=\dfrac{S^2} 4 \times \cot \phi.$$

Find the area of a regular decagon with side length $5.$

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A regular decagon is composed of $10$ congruent isosceles triangles. So the value of $\phi$ is

$$\phi=\dfrac{\pi}{n}=\dfrac{\pi}{10}.$$

Substituting this into the formula, we have

$$\text{Area of the decagon} =10\times \dfrac{S^2}{4} \times \cot \phi = 10\times \dfrac{5^2}{4} \times \cot \dfrac{\pi}{10} \approx 192.355.\ _\square$$

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Note: $\phi$ is in radian.