Chapter 13 — Polygons
Topics in this chapter:
13.1 Types of Polygons
A polygon is a simple closed planar figure formed by line segments.
Polygon Names
| # Sides | Name | # Sides | Name |
|---|---|---|---|
| 3 | Triangle | 8 | Octagon |
| 4 | Quadrilateral | 9 | Nonagon |
| 5 | Pentagon | 10 | Decagon |
| 6 | Hexagon | 12 | Dodecagon |
| 7 | Heptagon | $n$ | $n$-gon |
A regular polygon has all sides equal and all angles equal.
Warning. Equal sides alone do not make a polygon regular. Neither do equal angles alone (e.g. a non-square rectangle has equal angles but is not regular).
Diagonals
A diagonal is a segment from one vertex to a non-adjacent vertex.
Number of diagonals in an $n$-gon:
\[\frac{n(n-3)}{2}\]Derivation: Each of $n$ vertices connects to $n - 3$ others via diagonals. Since each diagonal is counted twice: $\frac{n(n-3)}{2}$.
13.2 Angles in a Polygon
Interior angle sum. For an $n$-sided polygon:
\[\text{Sum of interior angles} = 180°(n - 2)\]Proof: Draw $n - 3$ diagonals from one vertex, dividing the polygon into $n - 2$ triangles (each with angle sum $180°$).
Exterior angle sum. At each vertex the exterior angle $\alpha_i$ and interior angle $\beta_i$ satisfy $\alpha_i + \beta_i = 180°$. Summing over all $n$ vertices:
\[\sum \alpha_i + \sum \beta_i = 180n\]Since $\sum \beta_i = 180(n - 2)$:
\[\sum \alpha_i = 180n - 180(n - 2) = 360°\]The sum of the exterior angles of any convex polygon is $360°$.
13.3 Regular Polygons
For a regular $n$-gon, all interior angles and all exterior angles are equal.
Each interior angle $= \dfrac{180°(n-2)}{n}$
Each exterior angle $= \dfrac{360°}{n}$
Interior Angles of Common Regular Polygons
| # Sides | Interior Angle | # Sides | Interior Angle |
|---|---|---|---|
| 3 | $60°$ | 8 | $135°$ |
| 4 | $90°$ | 9 | $140°$ |
| 5 | $108°$ | 10 | $144°$ |
| 6 | $120°$ | 12 | $150°$ |
Inscribed and Circumscribed Circles
For any regular polygon, the perpendicular bisectors of the sides and the angle bisectors all meet at the center $O$. This gives:
- A circumscribed circle (circumradius $R$) passing through all vertices.
- An inscribed circle (inradius $r$, also called the apothem) tangent to all sides.
If the polygon has $n$ sides of length $l$, and $\theta = 180°/n$:
\[\tan\theta = \frac{l/2}{r}, \qquad \sin\theta = \frac{l/2}{R}\]Area of a regular polygon:
\[A = \frac{1}{2} n l r\]where $r$ is the apothem (inradius).
Example 13-1. Prove that connecting every other vertex of a regular hexagon forms an equilateral triangle.
Proof: By SAS: $\triangle ABC \cong \triangle CDE \cong \triangle EFA$ (two sides are sides of the hexagon, included angle $= 120°$). Thus $AC = CE = AE$, so $\triangle ACE$ is equilateral.
Example 13-2. Find the number of sides of a polygon whose interior angles sum to $2340°$.
Solution: $180(n - 2) = 2340 \implies n - 2 = 13 \implies n = 15$.
Example 13-3. Regular dodecagon $ABCD\cdots L$ with $AB = 4$. Find $AD$.
Solution: Each interior angle is $150°$. Trapezoid $ABCD$ is isosceles with $\angle BAX = 30°$. Drawing perpendiculars $BX$ and $CY$ to $AD$: $BCYX$ is a rectangle with $XY = 4$. In $\triangle ABX$: $BX = 2$, $AX = 2\sqrt{3}$. Similarly $YD = 2\sqrt{3}$.
\[AD = AX + XY + YD = 4 + 4\sqrt{3}\]Exercise 13-1. Find the number of sides of a regular polygon with interior angle $162°$.
Exercise 13-2. Prove that connecting every other vertex of a regular octagon forms a square. (Remember: showing sides equal is not enough — you must also show right angles.)
13.4 Regular Hexagons
Regular hexagons appear frequently because their numbers are relatively simple.
Drawing lines from the center to the vertices of a regular hexagon creates 6 equilateral triangles (each central angle is $60°$ and the two radii equal the side length).
Area of a regular hexagon with side $s$:
\[[ABCDEF] = 6 \cdot \frac{s^2\sqrt{3}}{4} = \frac{3s^2\sqrt{3}}{2}\]The longest diagonals (through the center) have length $2s$.
Example 13-4. Six points equally spaced on a circle of radius $1$. Find the total length of all possible segments.
Solution: The six points form a regular hexagon inscribed in the circle, so the side length is $1$.
- 6 sides of length $1$: total $= 6$.
- 3 long diagonals of length $2$: total $= 6$.
- 6 short diagonals (like $AC$): Since $\triangle ABC$ is isosceles with $\angle ABC = 120°$ and $AB = 1$, we get $\angle BAC = 30°$. Since $\angle FAB = 120°$, $\angle FAC = 90°$, making $\triangle CAF$ a $30°$-$60°$-$90°$ triangle with $AC = AF\sqrt{3} = \sqrt{3}$. Total $= 6\sqrt{3}$.
Grand total: $6 + 6 + 6\sqrt{3} = 12 + 6\sqrt{3}$.
Exercise 13-3. The shortest diagonal of a regular hexagon is $8\sqrt{3}$. What is the radius of the inscribed circle? (MAΘ 1990)
Exercises & Problems
218. Find the number of diagonals in a polygon of $100$ sides. (AHSME 1950)
219. $ABCDEF$ is a regular hexagon with side $6$. Find the area of $\triangle BCE$. (MATHCOUNTS 1986)
220. Two angles of a convex octagon are congruent. Each of the other six is triple each of the first two. Find the larger angle measure. (MAΘ 1990)
221. Two congruent regular $20$-sided polygons share a side. Find $\angle ACB$. (MATHCOUNTS 1992)
222. An equilateral triangle and a regular hexagon have equal perimeters. If the triangle’s area is $2$, find the hexagon’s area. (AHSME 1970)
223. Two coplanar regular hexagons share side $EF$. Given that the perimeter of quadrilateral $ABCD$ is $44 + 22\sqrt{3}$, find $EF$. (MATHCOUNTS 1992)
224. Find the area of a regular dodecagon inscribed in a circle of circumference $12\pi$. (MAΘ 1990)
225. Find the ratio of the area of a circle inscribed in a regular hexagon to the area of the circle circumscribed about the same hexagon.
226. Each interior angle of polygon $P$ is $7.5$ times the exterior angle at the same vertex. Find the sum $S$ of the interior angles. Is $P$ necessarily regular? (AHSME 1960)
227. A regular polygon with exactly $20$ diagonals is inscribed in a circle. The polygon’s area is $144\sqrt{2}$. Find the circle’s area. (MAΘ 1990)
228. Twelve equally spaced points on a circle. How many chords connecting pairs of points are longer than the radius but shorter than the diameter? (MATHCOUNTS 1989)
229. In regular polygon $ABCDE\cdots$, $\angle ACD = 120°$. How many sides? (MAΘ 1992)
230. Numbers $1, 2, \ldots, n$ are equally spaced on a circle’s rim. If $15$ is directly opposite $49$, find $n$. (MAΘ 1987)
231. A goat is tethered to a corner of a regular $n$-gon building. Both the side length and tether length equal $r$. Find the grazing area as a function of $r$ and $n$. (MAΘ 1992)
232. Exactly three interior angles of a convex polygon are obtuse. What is the maximum number of sides? (AHSME 1985)
233. A regular hexagonal park has side $2$ km. Starting at a corner, Alice walks $5$ km along the perimeter. How far is she from her start? (AHSME 1986)
234. If the sum of all angles except one of a convex polygon is $2190°$, how many sides does it have? (AHSME 1973)
235. Find the sum of angles $1, 2, 3, 4, 5$ in a five-pointed star. (MAΘ 1987)