Polygons: Circumscribed

Introduction

Relevance of the Topic

The study of circumscribed polygons is a fundamental step in the field of Geometry. This topic provides the basis for understanding many other complex geometric concepts, such as the Theory of Triangles or the characterization of internal and external angles. The ability to identify, describe, and work with circumscribed polygons is therefore an essential tool for Mathematics students, especially in the 1st year of High School, to deepen their understanding of Euclidean geometry.

Contextualization

Within the vast world of Geometry, circumscribed polygons lie at the intersection between Plane Geometry and Analytical Geometry, bringing with them powerful and classic concepts to Mathematics. We are talking about polygons inscribed in a circumference, which implies a relationship of dependence and interaction between the vertices of the polygons and the circumference in which they are inscribed. The ability to determine and work with circumscribed polygons not only deepens the understanding of Geometry but also enhances students' analytical skills.

Theoretical Development

Components

• Definition of Circumscribed Polygons: Circumscribed polygons are those that have all their vertices on the same circumference. The circumference that contains all the vertices of a circumscribed polygon is called a circumscribed or perimeter circumference.

• Property of the Center of the Circumscribed Circumference: In a circumscribed polygon, the center of the circumscribed circumference coincides with the point of intersection of the bisectors of the internal angles.

• Relation of Measures of Vertices and Angles: In a circumscribed polygon with n sides, the ratio between the measures of the internal angles and the measure of the central angles is always (n-2):n, where n is the number of sides of the polygon.

• Inscription Theorem: A right angle inscribed in a circumference is a semicircle, therefore measuring 90 degrees. This implies that a square, being a polygon with four right angles, can be easily circumscribed by a circumference.

Key Terms

• Perimeter Circumference: It is the circumference that contains all the vertices of a circumscribed polygon. The center of this circumference coincides with the center of the polygon.

• Bisector of an Angle: It is the line or ray that divides an angle into two equal parts. In the case of circumscribed polygons, the point of intersection of the bisectors will be the center of the circumscribed circumference.

• Internal Angle and Central Angle: An internal angle of a polygon is formed by two consecutive sides of the polygon. A central angle is formed by two line segments that depart from points on a circle and have a common endpoint. The measure of a central angle is always equal to the measure of the corresponding internal angle.

• Inscription Theorem: A theorem that establishes that a right angle inscribed in a circumference is a semicircle, therefore measuring 90 degrees. It is essential in understanding the inscription of a square in a circumference.

Examples and Cases

• Circumscribed Triangular Polygon: An equilateral triangle can be circumscribed to a circumference. In an equilateral triangle, all internal angles measure 60 degrees. Therefore, according to the relationship between the measures of the angles and the sides of a circumscribed polygon, each central angle measures 120 degrees.

• Circumscribed Quadrilateral Polygon: A square is an example of a circumscribed polygon. Each internal angle of a square measures 90 degrees, according to the definition of the square. Using the relationship between the measures of the internal and central angles, each central angle of the square measures 135 degrees. Therefore, the center of the circumscribed circumference of a square is equidistant from the four vertices of the square.

• Circumscribed Pentagonal Polygon: A regular pentagon can also be circumscribed in a circumference. If we divide the regular pentagon into central isosceles triangles (triangles whose equal sides are the radii of the circumscribed circumference), each internal angle of the central triangle would measure 36 degrees, and each central angle of the pentagon would measure 72 degrees. The center of the circumference would be the center of gravity of the regular pentagon.

These examples show how the relationship between polygons and circumferences can be explored and understood through the study of circumscribed polygons.

Detailed Summary

Key Points

• Characterization of Circumscribed Polygons: Circumscribed polygons are those that have all their vertices on the same circumference. This characteristic allows for specific mathematical relationships to exist between the angles and sides of the polygon and the circle that contains it.

• Inscription of Polygons in Circumferences: The inscription of a polygon in a circumference implies that the center of the polygon and the center of the circumference coincide. This allows the relationship between the internal angles of the polygon and the central angles to be established.

• Aspects of Angular Measures: Understanding the internal and central angles, as well as the use of the inscription theorem, contributes to the analysis of angular measures in circumscribed polygons. For example, the ratio between the measures of the internal angles and the measure of the central angles in a polygon with n sides is (n-2):n.

• Bisector Theorem: The application of the bisector theorem, which states that all bisectors of a circumscribed polygon meet at the center of the circumscribed circumference, is essential for understanding the relationship between the vertices of the polygon and the circumscribed circumference.

Conclusions

• The relationship between circumscribed polygons and the circumference that contains them provides a deeper understanding of geometric concepts, such as angles and the Theory of Triangles.

• The study of circumscribed polygons leads to the identification of patterns that can be applied to different polygons, improving students' spatial visualization and analytical reasoning skills.

• The properties of circumscribed polygons allow for the graphical representation of geometric shapes, making the interpretation and understanding of visual information in different contexts clearer.

Suggested Exercises

1. Circumscribed Hexagonal Polygon: Given a regular hexagon inscribed in a circumference, determine the measure of an internal angle of the hexagon and the measure of a central angle.

2. Circumscribed Heptagonal Polygon: If a heptagon is circumscribed to a circumference, determine the ratio between the measure of an internal angle of the heptagon and the measure of a central angle.

3. Circumscribed Octagonal Polygon: In a regular octagon circumscribed to a circumference, what is the measure of each internal angle? And the measure of each central angle?