Contextualization

Theoretical Introduction

Polygons are flat geometric shapes formed by three or more sides. When each of these sides is tangential to a circumference, that is, touching the circumference at only one point, we call it a circumscribed polygon. Circumscribed polygons present a depth of mathematical concepts that go far beyond the basics, and we will explore some of them in this project.

The study of circumscribed polygons involves understanding topics such as the distance between points, the concept of tangency, and the calculation of the area and perimeter of a polygon. It is important to understand that in a circumscribed polygon, the radius of the circumference is directly related to the length of the polygon's side.

Furthermore, by analyzing polygons with a large number of sides, we improve our understanding of what the number π (Pi) is, which is extremely important in many branches of mathematics. This approach to understanding π is known as the method of exhaustion, an ancient calculation technique used to find the area and volume of figures.

Contextualization

Mathematics, particularly geometry, plays a crucial role in the world we live in, and the knowledge of circumscribed polygons is no exception. This topic is employed in a variety of fields, including engineering, architecture, product design, and even in video games.

For example, space engineers use concepts of circumscribed polygons to optimize designs of spacecraft, which will have to withstand reentry into Earth's atmosphere. In architecture, the technique of building with geodesic domes, popularized by Buckminster Fuller, uses circumscribed polygons to create highly efficient and resistant structures.

Suggested research topics for further content exploration:

• Khan Academy - Polygons
• Brasil Escola - Inscribed and Circumscribed Regular Polygons
• Mundo Educação - Polygons

Practical Activity

Activity Title: Tangencing the Adventure

Project Objective:

This project aims to provide students with the opportunity to:

1. Deeply understand the concept of circumscribed polygons.
2. Apply geometry knowledge to solve real-world problems.
3. Integrate mathematics with design and creativity skills.
4. Develop socio-emotional skills such as teamwork, communication, and time management.

Detailed Project Description:

In this project, student groups will be challenged to plan and build a circumscribed polygonal dome structure using straws and tape. This activity will involve the principles of circumscribed polygons, the practical application of mathematical concepts, and the development of architectural design and creativity skills.

Required Materials:

• Straws
• Tape
• Ruler
• Calculator
• Paper and pen for calculations

Step by Step:

1. The first step of this project is the theoretical review of the concepts about circumscribed polygons. Students should research and make a presentation to the group on the topic. They must identify the properties of circumscribed polygons and learn how they apply in the real world.

2. Next, students will plan their dome. This planning should involve mathematical calculations to determine how many straws and how much tape will be needed, and the drawing of the dome on paper.

3. Subsequently, students will build the dome, using straws to represent the sides of the polygons and tape to fix them. They should use the knowledge learned about circumscribed polygons to ensure that all sides of the polygons are tangential to the imaginary circumference of the dome.

4. Finally, each group will present their dome to the class, explaining the mathematical concepts used, the challenges encountered, and how they overcame them.

Project Deliverables:

At the end of the project, students must deliver the physical dome, along with a written report detailing the creation process and the concepts learned.

• Introduction: The report should start with an introduction to the concept of circumscribed polygons, emphasizing their relevance in real-world applications and the purpose of this project.

• Development: The report should cover the theory addressed in the project, explain the activity in detail, and present and discuss the results obtained.

• Conclusions: The report should end with a conclusion summarizing its main points, explaining the learnings obtained, and drawing conclusions about the project.

• Bibliography: Finally, the report should contain the bibliographic references used in the research and execution of the project.

The report should demonstrate a clear understanding of the concept of circumscribed polygons, as well as the application of this concept in the construction of the dome. It should also reflect the students' ability to work in a team, solve problems, and think creatively.