Regular Polygons

A regular polygon is a polygon where all sides and all interior angles have the same measurement.

Note: The more sides a regular polygon has, the more it resembles a circle.

The apothem is a perpendicular line segment drawn from the centre of a regular polygon to the midpoint of any of its sides.

Note that there are as many apothems as there are sides in a regular polygon.

The triangles formed through decomposition are isosceles triangles whose height corresponds to the regular polygon’s apothem.

Polygon inscribed in a circle
To illustrate that these triangles are indeed isosceles, it is possible to inscribe a regular polygon in a circle. Notice that the sides of the triangles represented by the dotted lines are, in fact, the radii of the circle.

The radii of a circle are always equal, thus all of the triangles are isosceles. Moreover, since the base of these triangles corresponds to the sides of the regular polygon, we can determine that all of the triangles are congruent (isometric).

Since they are isosceles and congruent, the angle measurements of these triangles can be determined.

||\begin{gather}\text{Measure of the central angle}=\dfrac{360^\circ}{n}\\\text{where}\\ n:\text{number of sides of the regular polygon}\end{gather}||

What is the measure of the central angle of the following regular decagon?

1. Identify the full angle formed by the angles at the centre

2. Apply the formula
   ||\begin{align*}
   \color{green}{\text{Measurement of the central angle}} &= \displaystyle \frac{360^\circ}{n} \\
   &= \displaystyle \frac{360^\circ}{10} \\
   &= \color{green}{36^\circ}
   \end{align*}||

Given that the polygon below is regular, what are the angle measurements of the triangle?

1. Find the measurement of the central angle
   
   ||\begin{align*}
   \color{green}{\text{measure of central angle}} &= \displaystyle \frac{360^\circ}{10} \\
   &= \color{green}{36^\circ}
   \end{align*}||

2. Determine the other two angles’ measurements
   
   Since it is an isosceles triangle, ||\begin{align*}
   180^\circ &= 2 \times\color{fuchsia}{\text{angle measure}} + 1 \times \color{green}{\text{measure of central angle}}\\
   180^\circ &= 2 \times \color{fuchsia}{\text{angle measure}} + 1 \times \color{green}{36^\circ}\\​
   144^\circ &= 2 \times\color{fuchsia}{\text{angle measure}}\\​
   \color{fuchsia}{72^\circ} &= \color{fuchsia}{\text{angle measure}}
   \end{align*}||

The Regular Hexagon

To demonstrate that |\triangle{ABI}| is indeed an equilateral triangle, determine the measure of each of the angles.

1. Find the measure of |\color{fuchsia}{\angle AIB}| (central angle)
   ||\begin{align*}
   \color{fuchsia}{m\angle{AIB}} &= \displaystyle \frac{360^\circ}{6} \\
   &= \color{fuchsia}{60^\circ}
   \end{align*}||

2. Find the measure of |\color{red}{\angle IAB}| (half of an interior angle)
   ||\begin{align*}
   \color{red}{m\angle IAB} &= \displaystyle \frac{1}{2}\left( \frac{(n-2)\times 180^\circ}{n}\right) \\
   &= \displaystyle \frac{1}{2}\left(\frac{(6-2)\times 180^\circ}{6}\right) \\
   &= 60^\circ
   \end{align*}||

3. Find the measure of |\color{blue}{\angle ABI}|
   Since the sum of the interior angles of a triangle is |180^\circ|, use the following formula :
   ||\begin{align*}
   180^\circ &= \color{fuchsia}{m\angle AIB} + \color{red}{m\angle IAB} + \color{blue}{m\angle ABI}\\
   180 ^\circ &= \color{fuchsia}{60^\circ} + \color{red}{60^\circ} + \color{blue}{m\angle ABI} \\
   60^\circ &= \color{blue}{m\angle ABI}
   \end{align*}||

4. Interpret the answer
   Since the three angles have a measure |60^\circ,| then the triangle is equiangular, and therefore, equilateral.

||\begin{align}\text{Measure of an Interior Angle}&=\dfrac{\text{sum of the interior angles}}{\text{number of sides of the polygon}}\\&=\dfrac{(n - 2)\times180^\circ}{n}\end{align}||

||\begin{align} \text{Measure of an Interior Angle} &= \displaystyle \frac{(n-2) \times 180^\circ}{n} \\
&= \frac{(7-2) \times 180^\circ}{7} \\
&\approx 128.57^\circ \end{align}||

In this situation, all of the measures of the isosceles triangle |OAB| have been calculated. Notice that angle |\color{#ec0000}{ABC}| is formed by angles |\color{#ff55c3}{\angle{ABO}}| and |\color{#ff55c3}{\angle{OBC}},| which are congruent.

||\begin{align}\color{#ec0000}{\text{Measure of an Interior Angle}}&=2\times\color{#ff55c3}{m\angle{ABO}}\\&=2\times 72^\circ\\&=144^\circ\end{align}||

An exterior angle of a polygon is an angle formed between one side of the polygon and the extension of the adjacent side.

In the following figure, angle |ABE| is an exterior angle.

The sum of the exterior angles of any convex polygon is always 360°. The following animation shows the nine exterior angles of an enneagon (nonagon). Press the "play" button at the bottom left to start the animation.

||\begin{align}\text{Measurement of an Exterior Angle}&=\dfrac{360^\circ}{\text{number of sides of the polygon}}\\&=\dfrac{360^\circ}{n}\end{align}||

Here is the measure of an exterior angle of a regular pentagon.

||\begin{align}\text{Measure of an Exterior Angle}&=\dfrac{360^\circ}{n}\\
&=\dfrac{360^\circ}{5}\\
&=72^\circ\end{align}||

The interior angle and exterior angle of a polygon are supplementary. Using this information, the following two formulas are obtained:

||\begin{gather}\text{Measure of Exterior Angle}=180^\circ-\text{Measure of Interior Angle}\\\\\text{and}\\\\\text{Measure of Interior Angle}=180^\circ-\text{Measure of Exterior Angle}\end{gather}||

A regular polygon has as many axes of symmetry as there are sides. Thus, an equilateral triangle has three axes of symmetry and a regular hexagon has six.