Introduction

Relevance of the Theme

Classifying polygons is fundamental in the study of geometric shapes. This classification allows for the understanding of peculiar properties of different polygons, refining visual perception and the ability to correctly name these shapes. Such competence is crucial for solving more complex problems, based on the identification and application of geometric properties.

Contextualization

The study of polygons in Mathematics occurs within the broader context of Geometry, which is one of the main subdivisions within this science. Polygons are a specific type of closed two-dimensional figure that can be found in many aspects of everyday life, from house plans and city maps to graphic designs and digital games. The ability to classify polygons is one of the first steps in understanding how Geometry applies to the world around us. Furthermore, the classification of polygons is a fundamental skill for the study of more advanced topics, such as trigonometry and calculus.

Theoretical Development

Components

• Regular Polygons: Polygons that have all sides and angles equal. This characteristic of symmetry and regularity sets them apart from others. The most common examples are the square, the regular hexagon, and the regular octagon.

• Irregular Polygons: These are polygons that do not have equality between sides and angles. They have various shapes. Examples of irregular polygons are trapezoids, scalene triangles, and non-regular pentagons.

• Convex and Concave Polygons: The distinction between convex polygons (all internal angles are less than 180°) and concave polygons (at least one internal angle is greater than 180°) is essential to understand the possible geometric configurations that a polygon can have.

• Simple and Complex Polygons: Simple polygons do not have self-intersections, meaning no portion of their boundary crosses. Complex polygons, or star-shaped polygons, have intersections along their boundaries. This is an important concept for the classification of more complicated polygons.

Key Terms

• Polygon: A two-dimensional geometric figure composed of a closed sequence of line segments, which are the sides, where each endpoint of a segment is the start of the next.

• Side: Each of the line segments that form a polygon.

• Vertex: Point where two sides of a polygon meet.

• Internal Angle: Angle formed by two consecutive sides of a polygon.

• External Angle: Angle formed by one side of a polygon and a line that is an extension of the neighboring side.

• Diagonal: Line segment that joins two non-consecutive vertices of a polygon.

Examples and Cases

• In a square, all sides have the same length and all internal angles are 90°, classifying it as a regular and convex polygon.

• In a pentagon, the internal angles measure 108° and, if all sides had the same length, it would be a regular and convex polygon. However, because they do not, it is classified as an irregular and convex polygon.

• In a star-shaped hexagon, all internal angles are 120°. Due to self-intersection, it is classified as a star-shaped or complex, concave, and regular polygon.

Detailed Summary

Key Points:

• Nature of Polygons and Importance of Their Classification: Polygons are closed two-dimensional figures formed by line segments called sides. The ability to classify polygons is fundamental because it allows for the identification and application of specific properties to each group. This, in turn, enhances visual perception and problem-solving in geometry.

• Regular and Irregular Polygons: The main difference between regular and irregular polygons lies in the equality (or lack thereof) of their sides and angles. Regular polygons, such as the square, the regular hexagon, and the regular octagon, have all sides and angles equal. Irregular polygons, on the other hand, do not have this equality.

• Convex and Concave Polygons: The classification regarding the convexity and concavity of a polygon is related to the measurement of its internal angles. In convex polygons, all angles are less than 180°. In concave polygons, at least one angle is greater than 180°.

• Simple and Complex Polygons: This classification considers whether a polygon has self-intersections, meaning if there are parts of its boundary that cross. Simple polygons do not have self-intersections, while complex polygons do.

Conclusions:

• Correctly classifying a polygon is essential for the study of geometric problems and for understanding their intrinsic characteristics.

• Each type of polygon (regular, irregular, convex, concave, simple, complex) has its own unique properties that can be applied to solve different types of questions.

Exercises:

1. Classification of Polygons: Given a set of geometric figures, classify them according to the types of polygons learned in class: regular, irregular, convex, concave, simple, and complex. Justify your answers.

2. Properties of Regular Polygons: List the main characteristics of a regular polygon. Use examples of regular polygons to illustrate your answers.

3. Convex and Concave Polygons: Draw an example of each type of polygon (convex and concave) and explain the differences between them in terms of their internal angle measurements. Use the terminology learned to describe your figures.