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.