Comprehensive Guide to Measurement and Geometry: Polygons

Learning Objectives for Polygon Study

By the end of this study material, students are expected to be able to perform the following tasks:

Define a polygon.
Differentiate between various types of polygons, specifically convex and concave polygons.
Distinguish between regular and irregular polygons.
Classify polygons based on the number of sides they possess.
Draw a figure given specific parameters or names.

Definition and Fundamental Characteristics of Polygons

A polygon is defined as a two-dimensional geometric figure that consists of a finite number of sides. Key characteristics that define a polygon include:

Composition: The sides of a polygon are made of straight line segments.
Connectivity: These line segments are connected to each other end to end.
Closure: A polygon must be a closed figure.

Anatomy of a Polygon

Specific terms are used to describe the parts of a polygon:

Sides or Edges: These are the line segments that form the boundary of the polygon.
Vertex or Corner: This is the specific point where two line segments meet.
Angle: An angle is formed at each vertex where the sides intersect.

Quantitative Examples of Polygon Components

The following examples illustrate the relationship between sides and vertices in different polygons:

Example 1: A polygon with $3$ sides has exactly $3$ vertices.
Example 2: A polygon with $4$ sides has exactly $4$ vertices.
Example 3: A polygon with $6$ sides has exactly $6$ vertices.

Classification of Angles Based on Measure

Angles are quantified using degrees and are classified based on their measurements as follows:

Acute Angle: An angle whose measure is more than $0^{\circ}$ but less than $90^{\circ}$ .
Right Angle: An angle whose measure is exactly $90^{\circ}$ .
Obtuse Angle: An angle whose measure is more than $90^{\circ}$ but less than $180^{\circ}$ .
Straight Angle: An angle whose measure is exactly $180^{\circ}$ .
Reflex Angle: An angle whose measure is more than $180^{\circ}$ but less than $360^{\circ}$ .
Full Rotation Angle: An angle whose measure is exactly $360^{\circ}$ .

Interior and Exterior Angles

Polygons possess angles both inside and outside their boundaries:

Interior Angle: Any angle that is located on the inside of a polygon.
Exterior Angle: Any angle that is located on the outside of a polygon.
Measurement Tool: A protractor is the instrument used to measure an angle.

Differentiation Between Convex and Concave Polygons

Polygons can be categorized based on the measure of their interior angles:

Convex Polygons: A convex polygon has all of its interior angles measuring less than $180^{\circ}$ .
Concave Polygons: A concave polygon has at least one interior angle that measures more than $180^{\circ}$ .

Classification Case Studies: Convex vs. Concave

The following is a list of figures categorized by their convexity:

Figure 1: Concave Polygon
Figure 2: Convex Polygon
Figure 3: Concave Polygon
Figure 4: Convex Polygon
Figure 5: Concave Polygon
Figure 6: Convex Polygon
Figure 7: Concave Polygon
Figure 8: Convex Polygon
Figure 9: Concave Polygon
Figure 10: Concave Polygon
Figure 11: Convex Polygon
Figure 12: Convex Polygon

Regular vs. Irregular Polygons

Polygons are further classified by the equality of their sides and angles:

Regular Polygon: A polygon that has equal angles and equal sides, meaning all sides are exactly the same length.
Irregular Polygon: A polygon that has sides of varying sizes and angles of varying measures.

Irregular polygons may present in several ways:

Instances where angles are equal but sides are not equal.
Instances where sides are equal but angles are not equal.
Instances where neither the sides nor the angles are equal.

Types of Polygons According to the Number of Sides

Polygons are named based on the finite number of sides they contain:

Triangle: A polygon with $3$ sides.
Quadrilateral: A polygon with $4$ sides.
Pentagon: A polygon with $5$ sides.
Hexagon: A polygon with $6$ sides.
Heptagon: A polygon with $7$ sides.
Octagon: A polygon with $8$ sides.
Nonagon: A polygon with $9$ sides.
Decagon: A polygon with $10$ sides.
Undecagon: A polygon with $11$ sides.
Dodecagon: A polygon with $12$ sides.