Introduction

Key Theoretical Concepts

Inscribed and circumscribed figures are present in various aspects of our lives, and mathematics allows us to understand their structure and relationship. The idea of an inscribed polygon is that all its vertices are located on the circumference of a circle. On the other hand, a circumscribed polygon has all its sides tangent to the circumference of a circle.

Throughout history, these concepts have been widely used in art, architecture, and design. For example, in the 15th century, Leonardo da Vinci illustrated the 'Vitruvian Man', where a man with his arms and legs extended is inscribed in a circle and in a square. This shows the importance of these concepts in proportion and symmetry.

Analyzing the geometric relationship between the sides, apothems, and radii of triangles, squares, or hexagons that are inscribed or circumscribed in certain circles, in addition to facilitating the understanding of these shapes, can develop logical reasoning and problem-solving skills.

Contextualization

Through the study of inscribed and circumscribed figures, we can see how these concepts are present in our daily lives. In company logos, for example, it is common to find circles circumscribed or inscribed in triangles or squares. Furthermore, in architecture and product design, the notion of inscription and circumscription is frequently used to create symmetrical aesthetic patterns.

Its importance can also be noted in nature, where flowers, starfish, and many other organisms exhibit inscribed and circumscribed patterns, revealing the mathematical beauty present in the natural world. Understanding this logic allows us not only to appreciate the beauty of these forms but also to enhance the resolution of practical problems in various areas.

Practical Activity

Title: 'The Mathematics of Inscribed and Circumscribed: From Theory to Practice'

Project Objective

The objective of this project is to deepen the students' knowledge about inscribed and circumscribed polygons and their practical applications.

Detailed Project Description

In this project, students will investigate the geometric relationships of inscribed and circumscribed figures. They will construct a regular hexagon inscribed in a circle using only squares and a compass and, from this, they will identify the relationships between the sides, apothems, and radii of the hexagon and the circle. The activity will be carried out in groups of 3 to 5 students and will last for one month.

Necessary Materials

• Graph paper
• Pencil
• Eraser
• Ruler
• Compass
• Square

Step by Step

1. Research Stage: Students must conduct theoretical research on the topic, using the resources suggested in the introduction. In this stage, students must identify and understand the formulas related to inscribed and circumscribed polygons.

2. Construction Stage: Based on the research, students must build an algorithm for constructing a regular hexagon inscribed in a circle, using a compass and square. They must document this algorithm in writing and through a flowchart.

3. Execution Stage: Students must follow the algorithm to draw the regular hexagon inscribed in the circle on graph paper. Throughout the construction, students must identify the relationships between the sides, apothems, and radii of the hexagon and the circle.

4. Discussion Stage: After the construction, students must discuss the practical implications of this geometry. Examples can be sought in the fields of art, architecture, product design, etc.

5. Report Writing Stage: Finally, students must write a report detailing the experience. The report should be divided into four sections: Introduction, Development, Conclusions, and Bibliography. The introduction should describe the project's objective and its relevance. In Development, students should detail the algorithm for constructing the inscribed hexagon and the identified geometric relationships. In Conclusions, students should report what they learned from the activity and how it enhanced their understanding of the topic. Finally, in the Bibliography, students should list the sources used during the research and cite them correctly.

Project Delivery

The final project delivery includes two parts:

1. Diagram and Construction: The flowchart and construction of the inscribed hexagon, presenting the identified geometric relationships.

2. Written Report: A detailed report discussing the project's experience. This report should complement the practical part of the project, explaining the construction process, the observed mathematical relationships, and the relevance of these concepts in practical life.

Remember, organization, teamwork, and commitment to deadlines are as important as the correct execution of the activity.