Contextualization

Inscribed polygons are a fundamental part of geometry, and learning about them can do wonders to enhance your understanding of mathematics.

In a more technical language, a polygon is said to be inscribed in a circle when all its vertices are points on the circle. This means that we can draw a circle around the polygon in such a way that each of its corners touches the circle. Inscribed polygons are, therefore, an elegant way to connect plane geometry and circular geometry.

Now, you may wonder why an inscribed polygon is important? First, they occur in many places in the world around us. Nature, for example, uses geometric shapes for structure and function. Observe the structure of a snowflake or the arrangement of seeds in a sunflower; they are based on geometric patterns. Inscribed polygons, in particular, are used in engineering and architecture to help design complex shapes and structures, from the structure of a bridge to the design of an airplane wing.

Moreover, they are an important part of the study of mathematics and are a step towards understanding more complex concepts. In fact, many mathematical problems can be easily solved using inscribed polygons. Whether it is to calculate the area of an irregular shape or to determine the shortest distance between two points, inscribed polygons can be a valuable tool.

Practical Activity

Activity Title: Deciphering Inscribed Polygons

Project Objective

The objective of this project is to enhance students' knowledge, understanding, and skills in solving problems involving inscribed polygons. Students will have to draw inscribed polygons, perform calculations related to them, and finally write a report based on their observations and collected data.

Detailed Project Description

In this project, each group of 3 to 5 students will create a set of inscribed polygons using simple materials and then explore the underlying mathematical concepts. This will take about 3 hours and should be completed in one week.

The groups will draw, measure, and calculate the attributes of the inscribed polygons to explore the relationships between the sides of the polygons, their internal angles, and the radius of the inscribed circle.

Required Materials

1. Sheets of paper
2. Compass
3. Graduated ruler
4. Pencil and eraser
5. Calculator

Detailed Step-by-Step for Activity Execution

1. Drawing of inscribed polygons: Using the compass, each group must draw a circle with a radius of 5 cm on a sheet of paper. Next, the groups should draw inscribed polygons (triangle, square, pentagon, etc.) in this circle.

2. Measurement and calculations: Using the graduated ruler, the groups must measure the length of each side of the inscribed polygons. Next, they should calculate the perimeter (sum of the sides) and the value of each internal angle of the polygons.

3. Exploring relationships: The groups must explore the relationship between the side of an inscribed polygon and the radius of the circle. They should record their findings and explain any pattern or relationship they observe between the sides of the polygons, the internal angles, and the radius of the circle.

4. Writing the report: After completing the practical part, each group must prepare a report with the following topics: Introduction, Development, Conclusions, and Bibliography.

   • In the Introduction, students must contextualize the theme, its relevance, real-world applications, and the project's objective.

   • In the Development, students must explain the theory behind inscribed polygons, describe the activity in detail, include the methodology used, and present and discuss the results obtained.

   • In the Conclusions, students must summarize the main points, explain the learnings obtained, and draw conclusions about the project.

   • In the Bibliography, students must indicate the sources they relied on to work on the project, such as books, web pages, videos, etc.

Project Delivery

The project will be considered completed when students deliver:

• The drawings of the inscribed polygons.
• The complete report, which must fit and complement what they worked on in the practical activity.
• An oral presentation to the class telling a little about the work, its challenges, and the discoveries the group made.

Remember that the focus of this project is not only to assess students' knowledge of inscribed polygons but also to assess and develop skills such as teamwork, time management, and communication.