Introduction

Polygons are closed flat figures, composed of line segments. They emerged as a fundamental part of geometry, already in ancient Greece, with the studies of Euclid.

Polygons are classified as regular and irregular. A polygon is said to be regular when it has all sides and angles equal. For example, a square or a regular hexagon, which have respectively four and six sides of the same length.

Regular polygons also have interesting characteristics, such as the existence of diagonals that pass through the center of the figure, forming internal and external angles that can be calculated based on the number of sides that compose them. These characteristics allow these polygons to be used in various areas of knowledge, such as engineering, architecture, arts, etc.

Contextualization

Regular polygons are more present in our world than we imagine. They can be seen in nature, in the way bees build their hives in a hexagonal shape, or how snowflakes form perfect hexagonal structures.

These geometric figures are also frequently used in art, design, and architecture. In the Renaissance, artists and architects like Leonardo da Vinci and Brunelleschi used regular polygons to create their works of art and buildings.

In modern times, engineers and computer scientists use regular polygons in the development of 3D graphics for games and animations. Each three-dimensional object in a computer game or animation is composed of polygons, usually triangles.

To learn more about the concept of polygons and their characteristics, you can consult the following sources:

• Book: 'Mathematics and its technologies', by Claudio Marques (FTD Publisher)
• Website: Mathematics Portal (www.portaldamatematica.ufpr.br)
• Video: 'What are polygons?' available on the Youtube channel 'Matemática Rio with Prof. Rafael Procopio'

Practical Activity

Activity Title: Discovering the World of Regular Polygons

Project Objective:

The objective is to study the theory and properties of regular polygons, develop teamwork skills, problem-solving, and time management.

Detailed Description:

Students, in groups of 3 to 5, will explore the role of regular polygons in different fields of knowledge. Additionally, they will also calculate the number of diagonals passing through the center of the regular polygon and determine the internal and external angles of regular polygons using mathematical formulas.

Required Materials:

1. Pencil and paper
2. Ruler
3. Compass
4. Scientific calculator
5. Reference books
6. Internet access for research
7. Presentation materials (Poster paper, markers, etc.)

Detailed Step-by-Step:

1. The first part of the project involves theoretical research. Students should research regular polygons, their properties, how to calculate the number of diagonals passing through the center of the figure, and calculate the internal and external angles of regular polygons.

2. Next, each group should choose a field of knowledge (such as engineering, architecture, art, computer science, etc.) and research how regular polygons are used in that field. They should prepare a presentation to explain their findings.

3. The next step is the mathematical practice. Students should select some regular polygons and then calculate the number of diagonals passing through the center, as well as the internal and external angles. They should record their calculations, as well as any patterns or relationships they observe.

4. As a final step, each group should prepare a report. The report will include an introduction, theoretical explanation of regular polygons and their properties, a detailed description of how they made their calculations with the results obtained, and conclusions about what they learned.

Project Deliverables:

1. Presentation of the research on the application of regular polygons in a specific field of knowledge.

2. Record of mathematical calculations performed for the selected regular polygons, including the number of diagonals passing through the center and angle calculations.

3. Complete report including introduction, theoretical explanation, description of the methodology, presentation and discussion of the results, conclusions, and a list of the research sources used.

Students should work collaboratively to produce these deliverables. The project will help them improve their time management skills, as they have to divide tasks among group members to meet the deadline.

When writing the written document, it is essential for students to use the lessons learned during mathematical calculations and theoretical research to discuss the results. Discussions on how regular polygons are used in different fields should be based on the research conducted for the presentation, and the conclusion should connect the various parts of the project, reflecting on what was learned.