Polygons

Understanding Polygons

In geometry, polygons form a large and important family of shapes. In this chapter, we focus on what makes a polygon a polygon, how to describe and classify polygons in general, and on some basic properties that apply to all polygons (before looking specifically at quadrilaterals and regular polygons in their own chapters).

What Is a Polygon?

A polygon is a flat (two-dimensional) shape made from straight line segments that are joined end to end to form a closed path.

More precisely, a polygon:

• Lies in a plane (it is a 2D shape).
• Is made up only of straight line segments.
• Has segments that meet at endpoints called vertices.
• Has each segment intersect only with its immediate neighbors at its endpoints (no crossing over).
• Forms a closed figure: you can “walk” around it and return to your starting point without leaving the boundary.

Curved shapes or figures with gaps or crossings are not polygons.

Examples of polygons include triangles, quadrilaterals, pentagons, and hexagons.

Basic Terminology

For a polygon, we use several standard terms:

• Side: A straight line segment forming part of the boundary.
• Vertex (plural: vertices): A point where two sides meet.
• Interior: The region inside the polygon.
• Exterior: The region outside the polygon.
• Diagonal: A line segment joining two non-adjacent vertices (vertices that are not connected by a side).

If a polygon has $n$ sides, we call it an $n$-gon. Some common names:

• 3 sides: triangle
• 4 sides: quadrilateral
• 5 sides: pentagon
• 6 sides: hexagon
• 7 sides: heptagon
• 8 sides: octagon
• 9 sides: nonagon
• 10 sides: decagon

In general, “$n$-gon” is a convenient way to talk about a polygon with $n$ sides, especially when $n$ is large.

Counting Sides, Vertices, and Diagonals

For any polygon, the number of sides and the number of vertices are the same. If a polygon has $n$ sides, it has $n$ vertices.

Diagonals give a sense of how “complex” a polygon is. From a given vertex, you can draw diagonals to some of the other vertices.

To find the number of diagonals in an $n$-gon:

• From one vertex, you cannot draw a diagonal to itself or to its two adjacent vertices, but you can draw diagonals to the other $n - 3$ vertices.
• There are $n$ vertices, so starting this way would count $n(n-3)$ diagonals.
• Each diagonal has two endpoints, so it is counted twice in that product.

Therefore, the total number of diagonals in an $n$-gon is
$$
\text{Number of diagonals} = \frac{n(n - 3)}{2}.
$$

For example:

• A pentagon ($n = 5$): $\dfrac{5(5-3)}{2} = \dfrac{5 \cdot 2}{2} = 5$ diagonals.
• A hexagon ($n = 6$): $\dfrac{6(6-3)}{2} = \dfrac{6 \cdot 3}{2} = 9$ diagonals.

Convex and Concave Polygons

Polygons come in two main types based on how they “bend”:

• A polygon is convex if every interior angle is less than $180^\circ$, and every line segment between two interior points stays inside the polygon.
• A polygon is concave if at least one interior angle is greater than $180^\circ$, or equivalently, if at least one diagonal lies partly outside the polygon.

Another practical way to think about it:

• In a convex polygon, no “caves in” appear in the shape; it bulges outward.
• In a concave polygon, there is at least one “dent” or inward corner.

Most basic formulas (like the interior angle sum formula below) apply to any polygon, convex or concave, but many standard arguments are easier to visualize for convex polygons.

Simple and Complex Polygons

A simple polygon is one whose sides intersect only at their endpoints. Most polygons we study in basic geometry are simple.

A complex (or self-intersecting) polygon has sides that cross over each other in other places besides the vertices (for example, a star-shaped figure created by drawing a five-pointed star with one continuous line). In elementary geometry, we typically focus on simple polygons and do not apply the usual area and angle formulas to complex polygons.

Unless stated otherwise, “polygon” usually means a simple polygon.

Regular vs. Irregular Polygons

A polygon is regular if it is both:

• Equilateral: all sides have equal length.
• Equiangular: all interior angles are equal.

If a polygon does not have both of these properties, it is an irregular polygon.

Examples:

• A square is a regular quadrilateral: all four sides and all four angles are equal.
• An equilateral triangle is a regular triangle.
• A rectangle that is not a square is irregular: all angles are equal, but not all sides are equal.
• A rhombus that is not a square is irregular: all sides are equal, but angles are not all equal.

Regular polygons have especially nice and symmetric properties, and they will be explored further in the chapter devoted to regular polygons.

Interior Angles of a Polygon

The interior angles of a polygon are the angles inside the polygon at each vertex.

For a triangle (3 sides), the sum of the interior angles is
$$
180^\circ.
$$

For an $n$-sided polygon, the sum of the interior angles is
$$
\text{Sum of interior angles} = (n - 2) \times 180^\circ.
$$

This formula can be understood by dividing the polygon into triangles (for a convex polygon):

• From one vertex, draw diagonals to all non-adjacent vertices.
• This divides the polygon into $(n - 2)$ triangles.
• Each triangle has angle sum $180^\circ$, and the total sum of all those triangle angles is the sum of the interior angles of the polygon.

Examples:

• Pentagon ($n = 5$): sum $= (5 - 2) \cdot 180^\circ = 3 \cdot 180^\circ = 540^\circ$.
• Hexagon ($n = 6$): sum $= (6 - 2) \cdot 180^\circ = 4 \cdot 180^\circ = 720^\circ$.

For a regular $n$-gon, each interior angle is the same, so each interior angle has measure
$$
\text{Each interior angle} = \frac{(n - 2) \times 180^\circ}{n}.
$$

This is often used to identify or construct regular polygons.

Exterior Angles of a Polygon

An exterior angle of a polygon is formed by extending one side of the polygon at a vertex and measuring the angle between this extension and the adjacent side.

At each vertex of a convex polygon, the interior angle and its adjacent exterior angle are supplementary:
$$
\text{Interior angle} + \text{Exterior angle} = 180^\circ.
$$

If you take one exterior angle at each vertex and walk around the polygon, turning the same way each time, an important fact is:

$$
\text{Sum of one exterior angle at each vertex} = 360^\circ.
$$

This is true for any convex polygon, regardless of how many sides it has.

For a regular $n$-gon, all exterior angles (one at each vertex) are equal, so each exterior angle is
$$
\text{Each exterior angle} = \frac{360^\circ}{n}.
$$

This is the counterpart to the interior angle formula and is especially useful when working with regular polygons.

Naming and Notation

Polygons are often named by listing their vertices in order (either clockwise or counterclockwise). For example, a pentagon with vertices $A$, $B$, $C$, $D$, $E$ in order around the shape is written as:

$$
ABCDE.
$$

The order indicates which vertices are connected by sides:

• Sides: $\overline{AB}$, $\overline{BC}$, $\overline{CD}$, $\overline{DE}$, $\overline{EA}$.
• Diagonals: $\overline{AC}$, $\overline{AD}$, $\overline{BD}$, $\overline{BE}$, $\overline{CE}$.

We may also simply call it a “pentagon $ABCDE$” when it is clear from context.

Classifying Polygons by Number of Sides

Besides the specific names mentioned, it is common to classify polygons broadly by the number of sides:

• Triangle: 3 sides.
• Quadrilateral: 4 sides (studied in detail separately).
• $n$-gon: $n$ sides, where $n \ge 3$.

For many problems, the exact name (pentagon, hexagon, etc.) is less important than the number of sides $n$, because angle sums, diagonals, and other general properties depend directly on $n$.

Summary of Key General Formulas

For a simple polygon with $n$ sides ($n \ge 3$):

• Number of sides: $n$
• Number of vertices: $n$
• Number of diagonals:
  $$
  \frac{n(n - 3)}{2}
  $$
• Sum of interior angles:
  $$
  (n - 2) \times 180^\circ
  $$
• For a regular $n$-gon:
• Each interior angle:
  $$
  \frac{(n - 2) \times 180^\circ}{n}
  $$
• Each exterior angle (one at each vertex):
  $$
  \frac{360^\circ}{n}
  $$

These general ideas and formulas apply to all polygons and will be used repeatedly when studying specific families of polygons such as quadrilaterals and regular polygons.