Regular Polygons

You already know some regular polygons: the equilateral triangle is a regular triangle, and the square is a regular quadrilateral.

📐Definition — Regular Polygon

A polygon is called regular if all its sides are equal and all its angles are equal.

Properties of Regular Polygons

⚡Theorem — Theorem 7.1 — Convexity

Every regular polygon is convex.

⚡Theorem — Theorem 7.2 — Inscribed and Circumscribed Circles

Every regular polygon has both an inscribed circle and a circumscribed circle, and their centers coincide.

The common center is called the center of the regular polygon.

Central Angle and Radius Formulas

Consider a regular $n$ -gon with side length $a_n$ , circumradius $R_n$ , and inradius $r_n$ .

The angle $\angle AOB$ where $O$ is the center and $AB$ is a side is called the central angle:

$\angle AOB = \frac{360°}{n}$

Drawing the altitude $OM$ from the center to side $AB$ : $\angle AOM = \angle BOM = \dfrac{180°}{n}$ and $AM = \dfrac{a_n}{2}$ .

From right triangle $OMB$ :

$\boxed{R_n = \frac{a_n}{2\sin\dfrac{180°}{n}}}, \qquad \boxed{r_n = \frac{a_n}{2\operatorname{tg}\dfrac{180°}{n}}}$

Drag the slider to explore how the inscribed and circumscribed circles relate to the polygon as the number of sides increases:

n = 6 sides

n = 6R = 2.000r = 1.732a = 2.000central∠ = 60.0°interior∠ = 120.0°area = 10.392

Formulas for Common Polygons

$n$	Circumradius $R_n$	Inradius $r_n$
$3$ (triangle)	$R_3 = \dfrac{a\sqrt{3}}{3}$	$r_3 = \dfrac{a\sqrt{3}}{6}$
$4$ (square)	$R_4 = \dfrac{a\sqrt{2}}{2}$	$r_4 = \dfrac{a}{2}$
$6$ (hexagon)	$R_6 = a$	$r_6 = \dfrac{a\sqrt{3}}{2}$

Key fact: For a regular hexagon, the side equals the circumradius: $a_6 = R_6$ .

Constructing Regular Polygons

Regular hexagon: Starting from any point $M$ on a circle, mark off consecutive chords equal to the radius. This gives 6 vertices of a regular hexagon.

Regular square: Draw two perpendicular diameters $AC$ and $BD$ . The endpoints $A$ , $B$ , $C$ , $D$ are the vertices of a square.

Regular triangle: Connect alternate vertices of a regular hexagon.

If a regular $n$ -gon has been constructed, a regular $2n$ -gon can be obtained by finding the midpoints of all arcs between adjacent vertices and adding them as new vertices.

✎Example — Finding the Circumscribed Hexagon

A regular triangle with side $18$ cm is inscribed in a circle. Find the side of the regular hexagon circumscribed about the same circle.

Solution. Circumradius of the triangle: $R_3 = \dfrac{18\sqrt{3}}{3} = 6\sqrt{3}$ cm.

The inradius of the circumscribed hexagon equals $R_3 = 6\sqrt{3}$ cm.

Since $r_6 = \dfrac{b\sqrt{3}}{2}$ , where $b$ is the hexagon side: $b = \dfrac{2r_6}{\sqrt{3}} = \dfrac{2 \cdot 6\sqrt{3}}{\sqrt{3}} = 12$ cm. $\blacktriangleleft$

The Golden Ratio

In constructing a regular pentagon, the ratio of diagonal to side equals:

$\varphi = \frac{\sqrt{5}+1}{2} \approx 1.618$

This number, called the golden ratio, appears throughout mathematics, art, and nature.

✎Example — Regular Polygon with Given Angle

Does a regular polygon with interior angle $177°$ exist? If so, what type?

Solution. For a regular $n$ -gon, the sum of interior angles is $180°(n-2)$ , so each angle is $\dfrac{180°(n-2)}{n}$ .

Setting this equal to $177°$ : $180°(n-2) = 177°n \implies 360° = 3n \implies n = 120$ .

Answer: Yes — a regular 120-gon. $\blacktriangleleft$

Exercises

✏Exercise

A regular hexagon has a circumradius of $R = 10$ cm. Find its side length, inradius, and area.

✏Exercise

A circle of radius $6$ cm has a regular triangle inscribed in it. Find the side length of the triangle and the inradius of that triangle.

Hint: Use $R_3 = \dfrac{a\sqrt{3}}{3}$ to find $a$ , then $r_3 = \dfrac{a\sqrt{3}}{6}$ .

✏Exercise

The interior angle of a regular polygon is $150°$ . How many sides does it have?