Contextualization

Theoretical Introduction

Inscribed polygons are very interesting and important plane geometric figures in the study of mathematics. An inscribed polygon is one that is contained within a circumference, so that all its vertices are on the circumference. That is, a polygon is inscribed in a circumference if and only if each of its vertices is a point on the circumference.

However, to perceive the beauty hidden in the geometry of these figures, we need to delve into concepts such as reason, proportion, similarity of triangles, in addition to a certain prior understanding of circles and their properties.

Regarding the relationship between the side of the inscribed polygon and the radius of the circumference, it is possible to prove it geometrically using basic similarity and trigonometry theorems. Knowing how this relationship is established not only reinforces the understanding of the very concept of an inscribed polygon, but also expands knowledge in trigonometry, proportions, and problem solving.

Practical Contextualization

The question of inscribed polygons is not limited to theory, and its application can be found in several areas of everyday life. When we study inscribed polygons, we are also studying the way objects fit into space, this has practical applications in several areas such as architecture, design, engineering, and even in art.

For example, when engineers design gear cogs, they are dealing directly with inscribed polygons. The spatial organization in architecture, when planning a space, often uses the idea of inscribed polygons to optimize the use of space. In the history of art, many artists, such as the famous Leonardo Da Vinci, used the concept of inscribed polygons in their works.

We invite you, then, to embark on this exploratory journey through the world of inscribed polygons, with both theoretical and practical perspectives, in order to understand the beauty and usefulness of these mathematical concepts.

Practical Activity: "Inscribed Polygons: A Practical and Tangible Geometry"

Project Objective

The purpose of this project is to unravel the concept of inscribed polygons, through a more practical and playful approach. The group needs to understand, practice, apply, and teach the concepts learned.

Detailed description of the project

Students will be divided into groups of 3 to 5 members. Each group will have the mission to build physical models of inscribed polygons and demonstrate, in a practical way, how the length of the side of the polygon relates to the radius of the inscribed circumference. This should be narrated and recorded in a video of up to 15 minutes, which will be shared with the class in a "flipped classroom" format.

To make the video, the group should research about editing software and how to record a quality explainer video. In addition, the video must be accompanied by a PowerPoint presentation that will be shared with the class, so that everyone can follow the explanation and make notes.

In addition to the video, the groups must submit a detailed written report containing: Introduction, Development, Conclusions and Bibliography used. The report should elucidate the processes used during the project, from the creation of physical models to the recording and editing of the video.

Necessary materials

• Cardboard or plywood
• Rope or twine
• Ruler, set square, and compass
• Markers of various colors
• Camera or smartphone for recordings
• Computer with video editing software
• Presentation software (eg: Microsoft PowerPoint, Google Slides)

Detailed step-by-step to carry out the activity

1. After forming the groups, each group should study and research in detail about inscribed polygons, as well as about the relationship of the side of the polygon with the radius of the inscribed circumference.

2. Using the materials above, each group should build physical models of inscribed polygons (triangle, square, pentagon, hexagon and so on) in circumferences with the same radius.

3. After building the models, the groups should study and understand the relationship between the side of each polygon and the radius of the circumference.

4. The groups should then plan and structure the script for the video with the presentation of the models and the explanation of the theory.

5. Record and edit the video as planned.

6. Make a PowerPoint presentation to accompany the video.

7. Write the detailed report on the entire project execution process.

Project Deliverables

Each group must deliver:

1. Video: The explanatory video (maximum 15 minutes) where they demonstrate the relationship between the side of the inscribed polygons and the radius of the circumference, using the models they built. The video should be clear, well edited, and informative.

2. Presentation: A PowerPoint presentation that illustrates the explanation given in the video. This presentation will be made available to the rest of the class so that they can follow the video and make notes.

3. Written report: The report must have the following topics:

   • Introduction: Relevance and application of inscribed polygons in the real world and the objective of this project.
   • Development: The theory of inscribed polygons, the detailed description of the activity carried out, the methodology used and the results obtained.
   • Conclusion: Review the main points, explain the lessons learned and the conclusions about the project.
   • Bibliography: The sources used during the project.

Students should pay attention to the drafting of the report, ensuring that all the details of the project's elaboration are well explained, thus creating a document that complements the video and the presentation.