Introduction

Relevance of the Topic

• Inscribed and Circumscribed Polygons: Euclidean geometry is the foundation of the geometry we have learned since antiquity, and polygons are fundamental figures in this context. It is crucial to understand how they relate to inscribed and circumscribed circumferences, which helps to strengthen the overall understanding of two-dimensional geometry.

• Sides, Radius, and Apothem: These are crucial elements in the description of polygons. Working with these measurements allows us to describe and differentiate polygons from each other, and better understand their geometric characteristics.

Contextualization

• This topic is vital in the sequence of geometry content. It serves as a "link" between the study of circumferences (from the 7th grade) and the more in-depth study of regular polygons and their specific properties (to be studied later in the 8th grade).

• By understanding the properties of inscribed and circumscribed polygons, students begin to perceive the "hidden geometry" within these figures, which prepares them to understand more abstract geometry concepts in the future.

• In summary, the study of the sides, radii, and apothems of inscribed and circumscribed polygons is crucial for the development of spatial visualization skills, logical reasoning, and problem-solving, which are fundamental competencies in mathematics and many other disciplines.

Theoretical Development

Components

• Inscribed Polygons: A polygon is inscribed in a circumference when all its vertices touch the circumference.

  • In such polygons, the center of the circumference is the center of the polygon.
  • This implies that the radius of the circumference is equal to the distance between the center and any vertex of the polygon.

• Circumscribed Polygons: A polygon is circumscribed to a circumference when all its sides are tangent to the circumference.

  • In these polygons, the radius of the circumference is perpendicular to any side of the polygon and touches the side at its midpoint.

• Side of the Polygon: A polygon is characterized by its sides which are its most distinct parts.

  • All sides of an inscribed or circumscribed polygon are equal, as they are always equidistant from the center of the circumference.

• Radius of the Circumference: It is the line segment whose ends are the center of the circumference and any point on the circumference itself.

  • In the context of inscribed and circumscribed polygons, the radius is crucial, as it is equal to the side of the inscribed polygon and the distance from the center of the circumscribed polygon to any vertex.

• Apothem of the Polygon: The apothem is the distance between the center of a regular polygon and the midpoint of one of its sides.

  • In the case of a circumscribed polygon, the apothem is equal to the radius of the circumscribed circumference.
  • In the case of an inscribed polygon, the apothem is smaller than the radius of the inscribed circumference.

Key Terms

• Polygons: These are closed planar figures, composed of line segments called sides.

• Radius: It is the line segment that connects the center of a circumference to any point on the circumference itself.

• Apothem: It is the distance between the center of a geometric figure and the midpoint of one of its sides.

Examples and Cases

• For a circumscribed polygon, the radius of the circumference (or apothem of the polygon) is represented by the dashed red line that connects the center of the circumference to the vertex of the polygon.
• For an inscribed polygon, the radius of the circumference (or apothem of the polygon) is represented by the dashed red line that connects the center to the side of the polygon.

These examples and cases illustrate the importance of the concepts of sides, radii, and apothems in the description of inscribed and circumscribed polygons, and how these elements are interrelated.

Detailed Summary

Relevant Points

• Identification of Inscribed and Circumscribed Polygons: The first step is to recognize whether a polygon is inscribed or circumscribed to a circumference. This is a crucial step to understand the properties that will be studied.

• Relationship between the Radius and Sides of Polygons: The radius of the circumference that contains an inscribed polygon or circumscribes a polygon is equal to the side of the inscribed polygon or the distance from the center of the circumscribed polygon to any vertex.

• Apothem of the Circumscribed and Inscribed Polygon: The apothem of a polygon is the distance between the center of the polygon and the midpoint of one of its sides. In the case of a circumscribed polygon, the apothem is equal to the radius of the circumscribed circumference. In the case of an inscribed polygon, the apothem is smaller than the radius of the inscribed circumference.

Conclusions

• Equidistance: All sides of an inscribed or circumscribed polygon are equidistant from the center of the circumference. This is true for both regular and irregular polygons.

• Similarity of Measurements: The equality between the radius and the side of the inscribed polygon, and between the radius and the apothem of the circumscribed polygon, is a very interesting relationship. It suggests a series of properties and applications extremely valuable in solving various geometric problems.

• Relevance of the Context: Understanding the concepts of inscribed and circumscribed polygons, radius and sides, and the relationship between them, is crucial for deepening the study of Euclidean geometry and preparing students for more advanced topics such as proportion and similarity of figures.

Suggested Exercises

1. Polygon Identification: Given a polygon and a circumference, identify whether the polygon is inscribed or circumscribed to the circumference. Justify your answer.

2. Determination of the Radius: For a polygon inscribed in a circumference with a radius measuring 5 units, what is the measure of the side of the polygon? Justify your answer.

3. Analysis of Apothems: In a polygon circumscribed to a circumference with a radius of 8 units, what is the distance from the center of the polygon to one of its vertices? And the distance from the center to one of its middle sides? Justify your answers.