Context
Geometry is a branch of mathematics that studies space and the figures that can occupy it. Within this field, inscribed and circumscribed figures are fundamental concepts. They are often used to solve complex problems in a simple way. A figure is said to be inscribed when all its vertices are on the circumference, that is, they are 'contained' within the circle. On the other hand, a figure is said to be circumscribed when the circumference is contained within the figure, that is, all sides of the figure are tangent to the circumference.
Not only in mathematics, the concept of inscribed and circumscribed figures is relevant in many areas. For example, in civil engineering and architecture, calculations involving the construction of complex geometric structures often use these properties to optimize the use of materials and labor. Therefore, understanding these concepts is crucial for solving real problems.
To familiarize yourself with these concepts, I suggest consulting the book 'Elements of Euclid,' which is a classic reference in geometry. Additionally, I recommend watching the video 'Inscribed and Circumscribed' available on YouTube and reading the article 'What are inscribed and circumscribed figures?' on the Toda Matéria website.
These resources will provide a solid foundation for understanding these geometric figures and their properties, as well as presenting practical examples that illustrate the applicability of these concepts in different contexts.
The goal of our project is not only to understand the theory behind inscribed and circumscribed figures but also to apply them in practice to solve problems. It will be a challenge, but I am sure you are capable of overcoming it!
Practical Activity: 'Exploring Inscribed and Circumscribed Figures'
Project Objective
The objective of this project is to give students the opportunity to expand their knowledge and skills in geometry, specifically about inscribed and circumscribed figures. Students will have to design, create, and analyze inscribed and circumscribed geometric figures, as well as write a detailed report on their exploration and findings.
Detailed Project Description
Students will be divided into groups of 3 to 5 people. Each group will be tasked with creating, using colored papers, a circle with an inscribed and circumscribed triangle, square, and hexagon. Each figure will have a different size, chosen by the group, and the radius measurement will be used to calculate all relevant properties.
Students will also need to calculate and note the geometric relationships between the sides, apothems, and radii of these figures. Additionally, they will have to create an algorithm for constructing a regular hexagon from the central angle measurement, using squares and a compass.
This task will require the application of geometry concepts, spatial reasoning skills, and will also develop socio-emotional skills such as collaboration, time management, critical thinking, and problem-solving.
Required Materials
- Colored Cardstock Paper
- Ruler
- Compass
- Square
- Colored Pens
- Calculator
- Notebook
Detailed Step-by-Step for Activity Execution
- Divide the class into groups of 3 to 5 students.
- Each group should choose three different sizes of radii for the circles, one for each figure (triangle, square, and hexagon).
- The group should draw the circles on the cardstock paper using the compass.
- Next, each group should draw the inscribed and circumscribed geometric figures (triangle, square, and hexagon) in the circles, using the square and ruler.
- Each group should record all measurements in the notebook and calculate the relevant geometric relationships (lengths of sides, apothems, radii, areas, perimeters, etc).
- Students should use the central angle measurement to create an algorithm for constructing a regular hexagon, using squares and a compass.
- Finally, each group should write a report on the project, including information on the calculations performed, observations made, and conclusions drawn.
Upon completion of the project, students should submit the report and the produced geometric figures. The report should include an introduction, development, conclusions, and bibliography used. In the introduction, students should contextualize the theme and the project's objective. In the development, they should explain the theory behind inscribed and circumscribed figures, detail the calculations performed, the methodology used, and discuss the results obtained. In the conclusion, they should summarize the main points of the project, the learnings obtained, and the lessons learned. Finally, in the bibliography, they should indicate the sources used.
