Polygons: Definition and Classification

Polygons: Introduction to Polygons

In geometry, we progress from basic elements like points, lines, and planes to combining these elements to form more complex figures. One of the fundamental classes of closed two-dimensional shapes is the polygon. Polygons are essentially plane figures constructed from a finite sequence of straight line segments that connect to form a closed loop without intersecting themselves (except at the vertices).

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Definition

A polygon is a closed plane figure formed by a finite sequence of three or more straight line segments. These line segments are called the sides or edges of the polygon, and the points where the segments connect are called the vertices (singular: vertex). A polygon satisfies the following conditions:

1. It is a plane figure, meaning it lies entirely within a single plane.
2. It is formed by a finite number of line segments (at least three).
3. The segments are connected end-to-end, forming a closed path.
4. Each segment (side) intersects exactly two other segments, one at each of its endpoints (vertices).
5. The segments (sides) do not intersect each other except at their endpoints (vertices). This describes a simple polygon (as opposed to self-intersecting polygons).

Here are some examples of polygons:

The term "polygon" comes from the Greek words "poly" (meaning "many") and "gon" (meaning "angle"). Since a polygon has vertices, it also has interior angles formed at each vertex.

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Essential Properties and Non-Examples

Polygons possess several basic characteristics:

• They are strictly two-dimensional figures, lying flat on a plane.
• They are bounded by straight line segments. Curved boundaries are not allowed in a polygon.
• They must be closed figures; there should be no openings or gaps in the boundary.
• For simple polygons, the sides do not cross over each other between the vertices.

The simplest polygon is the triangle, which has the minimum required number of sides (three). Figures with fewer than three line segments cannot form a closed, bounded region in a plane.

It's helpful to understand what is not a polygon. Figures that fail one or more of the definition criteria are not polygons. Examples of non-polygons include:

From left to right, these examples are not polygons because:

• The first figure (circle) has a curved boundary, not line segments.
• The second figure (wavy line) is not closed; it has distinct start and end points.
• The third figure (star shape) is formed by line segments and is closed, but the segments intersect each other at points other than their endpoints. This is a polygon, but it's a complex or self-intersecting polygon, not a simple polygon as primarily studied in basic geometry. The common definition of "polygon" usually implies a simple polygon unless otherwise specified.

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In the following sections, we will explore specific terms related to polygons, their classification based on the number of sides and angle properties, and delve deeper into particular types like triangles and quadrilaterals.

Polygons: Terms Related to Polygons (Sides, Vertices, Diagonals, Interior/Exterior Angles)

To describe and analyse polygons effectively, we use specific terminology for their components and associated angles. Understanding these terms is fundamental to studying the properties and classifications of different polygons.

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Key Terms Related to Polygons

1. Sides

The sides (or edges) of a polygon are the straight line segments that form its boundary. These segments are connected end-to-end to create the closed figure. The number of sides is a crucial characteristic that determines the polygon's basic name (e.g., a triangle has 3 sides, a quadrilateral has 4 sides).

In the pentagon ABCDE shown, the sides are $\overline{\text{AB}}$, $\overline{\text{BC}}$, $\overline{\text{CD}}$, $\overline{\text{DE}}$, and $\overline{\text{EA}}$.

2. Vertices

The vertices (singular: vertex) of a polygon are the points where two sides meet or intersect. These are the "corners" of the polygon. Each vertex is shared by exactly two sides.

A polygon always has the same number of vertices as it has sides.

In the pentagon ABCDE, the vertices are A, B, C, D, and E.

3. Adjacent Sides

Adjacent sides are two sides of a polygon that are next to each other and share a common vertex.

In the pentagon ABCDE, $\overline{\text{AB}}$ and $\overline{\text{BC}}$ are adjacent sides because they share the common vertex B. Other pairs of adjacent sides include $\overline{\text{BC}}$ and $\overline{\text{CD}}$, $\overline{\text{CD}}$ and $\overline{\text{DE}}$, $\overline{\text{DE}}$ and $\overline{\text{EA}}$, and $\overline{\text{EA}}$ and $\overline{\text{AB}}$.

4. Adjacent Vertices

Adjacent vertices are two vertices of a polygon that are connected by a single side of the polygon. In other words, they are consecutive vertices along the boundary of the polygon.

In the pentagon ABCDE, A and B are adjacent vertices because they are connected by the side $\overline{\text{AB}}$. Other pairs of adjacent vertices include B and C, C and D, D and E, and E and A.

5. Diagonals

A diagonal of a polygon is a line segment that connects two non-adjacent vertices.

For a polygon with $n$ sides (and thus $n$ vertices), the number of diagonals can be calculated using a specific formula. From each vertex, you can draw a line segment to every other vertex except itself and its two adjacent vertices ($n - 3$ vertices). Since each diagonal connects two vertices, we divide by 2 to avoid counting each diagonal twice.

Formula for the number of diagonals:

For a polygon with $n$ sides, the number of diagonals is $\frac{n(n-3)}{2}$.

In the pentagon ABCDE ($n=5$), the diagonals are $\overline{\text{AC}}$, $\overline{\text{AD}}$, $\overline{\text{BD}}$, $\overline{\text{BE}}$, and $\overline{\text{CE}}$. Let's verify the formula:

The formula correctly gives 5 diagonals for a pentagon, which matches the diagram.

6. Interior Angles

An interior angle of a polygon is the angle formed inside the polygon at each vertex by the two adjacent sides that meet at that vertex. Every simple polygon has exactly one interior angle at each vertex.

In the pentagon ABCDE, $\angle \text{ABC}$, $\angle \text{BCD}$, $\angle \text{CDE}$, $\angle \text{DEA}$, and $\angle \text{EAB}$ are the interior angles.

7. Exterior Angles

An exterior angle of a polygon is formed at a vertex when one side is extended outwards. It is the angle between one side of the polygon and the extension of the adjacent side, lying outside the polygon. At each vertex of a convex polygon, the interior angle and its corresponding exterior angle form a linear pair.

In the figure, by extending side $\overline{\text{DC}}$ to point F, the angle $\angle \text{BCF}$ is an exterior angle at vertex C. Note that $\angle \text{BCD}$ (the interior angle at C) and $\angle \text{BCF}$ form a linear pair on the line DF, so $\text{m}\angle \text{BCD} + \text{m}\angle \text{BCF} = 180^\circ$.

At each vertex, there are actually two pairs of vertically opposite exterior angles, but we typically refer to just one at each vertex when discussing the sum of exterior angles of a polygon.

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Polygons: Classification based on Sides (Triangle, Quadrilateral, Pentagon, etc.)

One of the primary ways to classify polygons is by the number of line segments (sides) they have. Since a polygon must have at least three sides to form a closed figure, the number of sides is always a positive integer $n$ where $n \ge 3$. Each number of sides corresponds to a specific name for the polygon.

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Names of Polygons by Number of Sides

The names of polygons are typically derived from Greek or Latin prefixes indicating the number of sides. Here is a list of common polygon names based on their number of sides:

Number of Sides ($n$)	Name of Polygon	Common Name
3	Trigon	Triangle
4	Tetragon	Quadrilateral
5	Pentagon
6	Hexagon
7	Heptagon (or Septagon)
8	Octagon
9	Nonagon (or Enneagon)
10	Decagon
11	Hendecagon (or Undecagon)
12	Dodecagon
13	Tridecagon
14	Tetradecagon
15	Pentadecagon
16	Hexadecagon
17	Heptadecagon
18	Octadecagon
19	Enneadecagon
20	Icosagon
...	...	...
$n$ (for large $n$)	$n$-gon

For polygons with a large number of sides, especially those not listed above, it is common and acceptable to refer to them simply as an "$n$-gon", where $n$ is the number of sides (e.g., a 17-gon is a polygon with 17 sides).

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Properties Related to the Number of Sides

The number of sides of a polygon ($n$) is directly related to several other properties, such as the number of vertices, angles, and diagonals, as well as the sum of its interior and exterior angles. For any simple polygon with $n$ sides:

• Number of Vertices: A polygon always has the same number of vertices as it has sides. So, Number of Vertices $= n$.
• Number of Interior Angles: There is one interior angle at each vertex. So, Number of Interior Angles $= n$.
• Number of Exterior Angles: There is one exterior angle at each vertex (formed by one side and the extension of the adjacent side). So, Number of Exterior Angles $= n$.
• Sum of the Measures of the Interior Angles: The sum of the measures of the interior angles of any simple $n$-sided polygon can be calculated by dividing the polygon into triangles using diagonals from one vertex. From one vertex, $n-3$ diagonals can be drawn, dividing the polygon into $n-2$ triangles. Since the sum of angles in each triangle is $180^\circ$, the total sum of the interior angles of the polygon is $(n-2)$ times $180^\circ$.

  $\text{Sum of Interior Angles} = (n-2) \times 180^\circ$

• Sum of the Measures of the Exterior Angles: For any convex polygon, the sum of the measures of the exterior angles (taking one at each vertex) is always $360^\circ$. This property holds true regardless of the number of sides of the polygon.

  $\text{Sum of Exterior Angles} = 360^\circ$

• Number of Diagonals: As discussed in the previous section, the number of diagonals in an $n$-sided polygon is given by the formula:

  $\text{Number of Diagonals} = \frac{n(n-3)}{2}$

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Polygons: Types of Polygons (Convex, Concave, Regular, Irregular)

In addition to classifying polygons by the number of sides (like triangles, quadrilaterals, pentagons, etc.), we can also classify them based on the properties of their interior angles and the equality of their sides and angles. These classifications help us understand the overall shape and symmetry of a polygon.

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Classification Based on Interior Angles

Polygons can be divided into two major categories based on the measures of their interior angles:

1. Convex Polygon

A polygon is called a convex polygon if all of its interior angles are less than $180^\circ$. In a convex polygon, no interior angle is a reflex angle. Another way to think about it is that for any side of the polygon, the entire rest of the polygon lies on the same side of the line containing that side. Furthermore, all diagonals of a convex polygon lie entirely inside the polygon's boundary.

The polygons commonly encountered in basic geometry (triangles, squares, regular polygons, etc.) are generally convex. They do not have any "indentations" or "dents".

2. Concave Polygon (or Non-convex Polygon)

A polygon is called a concave polygon (or a non-convex polygon) if it is not convex. This means that a concave polygon has at least one interior angle that is a reflex angle (i.e., its measure is greater than $180^\circ$). A concave polygon has at least one "dent" or "cave" in its boundary. In a concave polygon, at least one diagonal lies partly or entirely outside the polygon.

The arrowhead shape (a quadrilateral) and the star shape shown are examples of concave polygons. The star shape can be viewed as a concave decagon or a self-intersecting pentagon depending on the definition used. In simple concave polygons, there's always at least one interior angle greater than $180^\circ$.

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Classification Based on Sides and Angles

Polygons are also classified based on the relationships between the lengths of their sides and the measures of their interior angles:

1. Regular Polygon

A polygon is called a regular polygon if it is both equilateral and equiangular.

• Equilateral: All sides of the polygon are equal in length.
• Equiangular: All interior angles of the polygon are equal in measure.

For a polygon to be regular, it must satisfy *both* conditions. For triangles, being equilateral implies being equiangular, and vice versa. However, for quadrilaterals and polygons with more sides, this is not always true (e.g., a rhombus is equilateral but not always equiangular; a rectangle is equiangular but not always equilateral).

Examples of regular polygons:

• Equilateral Triangle (3 equal sides, 3 equal angles of $60^\circ$)
• Square (4 equal sides, 4 equal angles of $90^\circ$)
• Regular Pentagon (5 equal sides, 5 equal angles)
• Regular Hexagon (6 equal sides, 6 equal angles)
• ... and so on for any number of sides $n \ge 3$.

All regular polygons are convex.

Properties of Regular Polygons:

For a regular polygon with $n$ sides:

• All $n$ sides are equal in length.
• All $n$ interior angles are equal in measure.
• All $n$ exterior angles are equal in measure.
• The measure of each interior angle is given by the formula:

  $\text{Each Interior Angle} = \frac{\text{Sum of Interior Angles}}{n} = \frac{(n-2) \times 180^\circ}{n}$

• The measure of each exterior angle is given by the formula:

  $\text{Each Exterior Angle} = \frac{\text{Sum of Exterior Angles}}{n} = \frac{360^\circ}{n}$

• The sum of an interior angle and its adjacent exterior angle at any vertex is always $180^\circ$.

2. Irregular Polygon

An irregular polygon is any polygon that is not regular. This means it is either not equilateral (sides are not all equal), or not equiangular (interior angles are not all equal), or neither.

Examples of irregular polygons:

• Scalene Triangle (all sides different, all angles different)
• Isosceles Triangle (two sides equal, two angles equal, unless it's equilateral)
• Rectangle (all angles are $90^\circ$, but sides are not necessarily equal)
• Rhombus (all sides are equal, but angles are not necessarily equal)
• Trapezium, Kite, etc.
• Most polygons drawn randomly.

Irregular polygons can be either convex or concave.

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