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 \(5.\)
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.
Challenge Problems
The figure below is a regular octagon.
Which area is larger, green or yellow?
The centers of the two regular hexagons shown below coincide.
The ratio of the area of the yellow region to the area of the green region is \(3:1.\)
What is the ratio of the perimeter of the big hexagon to the perimeter of the small hexagon?
The area of the yellow region in the regular five-pointed star shown below is \(500.\)
Find the area of the blue region rounded to the nearest integer.