Introduction to Polygons: Unveiling Geometric Shapes

Relevance of the Topic

Demystifying polygons, revealing that they are present in our daily lives through constructions, objects, and even in visual arts. These figures, besides composing the basis for more advanced mathematical studies, are the essence of geometry, a science that describes and analyzes the shapes and sizes of objects in space.

Contextualization

The study of polygons fits into the geometry component of the Mathematics curriculum in Elementary School. It is an interdisciplinary concept, serving as a foundation for more complex topics such as trigonometry and spatial geometry. Understanding polygons contributes to the development of logical and abstract thinking, as well as promoting spatial visualization skills and the ability to generalize and apply concepts.

Theoretical Development

Components of Polygons

• Sides: Line segments that form the outline of the polygon. Two consecutive sides always meet at a point, called a vertex. The number of sides defines the type of polygon.
• Vertices: Points where the sides of the polygon meet. Each vertex is shared by two sides.
• Interior Angles: Angles formed inside the polygon, at each vertex. When calculating these angles, the sum will always be 180°, regardless of the number of sides of the polygon. Formula for the sum of interior angles: (n-2)x180°, where n is the number of sides of the polygon.
• Diagonals: Line segments that connect non-consecutive vertices. The number of diagonals in a polygon depends on the number of sides and can be calculated by the expression: n(n-3)/2, where n is the number of sides.

Key Terms

• Polygon: Flat and closed figure, formed only by line segments called sides.
• Convex Polygon: Every point on the line segment that joins two points of the polygon belongs to its interior or its boundary.
• Concave Polygon: There is at least one point on the line segment that joins two points of the polygon that does not belong to its interior or its boundary.
• Regular Polygon: Convex polygon with all sides and angles equal.

Examples and Cases

• Triangle: Polygon with 3 sides and 3 interior angles that sum up to 180°. The number of diagonals is 0.
• Quadrilateral: Polygon with 4 sides and 4 interior angles that sum up to 360°. The number of diagonals is 2.
• Pentagon: Polygon with 5 sides and 5 interior angles that sum up to 540°. The number of diagonals is 5.
• Hexagon: Polygon with 6 sides and 6 interior angles that sum up to 720°. The number of diagonals is 9.
• Heptagon: Polygon with 7 sides and 7 interior angles that sum up to 900°. The number of diagonals is 14.

Detailed Summary

Key Points:

• Polygons are closed flat figures composed of line segments called sides, where each side meets two other sides at a point called a vertex.
• The sum of the interior angles of a polygon is always equal to 180° x (n-2), where n is the number of sides. This is because each vertex in a polygon contributes to the formation of an interior angle.
• The diagonals in a polygon are line segments that connect two non-consecutive vertices. The number of diagonals in a polygon can be calculated using the formula n(n-3)/2, where n is the number of sides.
• There are different types of polygons, including convex polygons (all points on the line segment that joins two points of the polygon belong to its interior or its boundary), concave polygons (there is at least one point on the line segment that joins two points of the polygon that does not belong to its interior or its boundary), and regular polygons (all sides and angles are equal).

Conclusions:

• Understanding the basic concepts of polygons, including the definition of sides, vertices, interior angles, and diagonals, is essential to explore more advanced topics in geometry, such as trigonometry and spatial geometry.
• Polygons are not just a mathematical abstraction; they are tangibly present in our daily environment, from the shapes of constructions to visual art.
• Manipulating and studying polygons promote cognitive skills, such as logical thinking, spatial visualization, and the ability to generalize and apply concepts.

Exercises:

1. Calculate the sum of the interior angles and the number of diagonals of an octagon (a polygon with 8 sides).
2. Identify whether each of the following figures is a convex, concave, or regular polygon: (a) square, (b) rhombus, (c) trapezoid, (d) pentagon.
3. Draw a polygon with 6 sides, calculate its sum of interior angles, and the number of diagonals.