Context

Introduction to the Topic

Geometry is a branch of mathematics that deals with the properties of space. In particular, Euclidean geometry is centered on the concepts of point, line, and plane, and the various shapes that can be constructed from these. Among these various shapes, one of the most fundamental and intriguing objects is the pyramid.

The pyramid is a solid figure formed by joining a polygon in the plane with a point outside that plane. This polygon is called the base, and the lines connecting the base to the point are called lateral edges. The point is the vertex of the pyramid. When we talk about pyramids, we often think of the great pyramids of Egypt, but they are much more than archaeological monuments. They are an object of mathematical study with several interesting properties to be investigated.

They are classified in geometry as solids with a polygonal base, which can be: quadrangular, triangular, pentagonal, among others. Pyramids also have an important characteristic: their lateral faces are triangles. We will see that these properties will be fundamental when evaluating the areas of a pyramid.

Relevance of Pyramids

The study of pyramids is very relevant in practice. Consider, for example, the construction of pyramid-shaped roofs for houses and buildings. An architect needs to calculate the area of the roof surface to know how much material will be needed for its construction. In addition, in engineering, pyramids help in calculating earthwork volumes. In biology, the concept of pyramids is used to represent the energy available at each level of a food chain. These are just a few of the numerous applications of pyramids in various disciplines.

In addition to their practical applications, the study of pyramids also helps develop abstract thinking and problem-solving skills, essential skills in many careers and daily life.

Practical Activity

Activity Title:

"Real World Pyramids: A Practical Study of Areas"

Project Objective:

This project aims to provide a practical and in-depth understanding of how to calculate the lateral surface area of a pyramid and the total surface area of a pyramid through a hands-on activity. Additionally, it aims to stimulate teamwork, organization, communication, and critical thinking.

Detailed Project Description:

Students will be divided into groups of 3 to 5 people. Each group must build a model of a pyramid, whose base can be any regular polygon, such as a triangle, square, pentagon, etc. After building the pyramid, the group must calculate the lateral surface area and the total surface area of the pyramid. To make the project more dynamic and challenging, the group must choose an appropriate scale (such as 1:50, 1:100, etc.) that will represent the proportion between the dimensions of the model and the dimensions of the pyramid if it were built in real life. Finally, students should research and write a report on real-world applications of pyramid area calculations. The document should be consistent with the chosen scale.

The project timeline will be 1 month, and the estimated workload is 5 to 10 hours per student, divided between creating the model, calculations, and report writing.

Required Materials:

• Cardboard or cardstock
• Tape measure/ruler
• Compass
• Pens, pencils, and eraser
• Scissors
• Glue
• Colored pens or markers
• Computer with internet access (for research)
• Word processing program (Microsoft Word, Google Docs, or similar)

Step-by-Step Guide for the Activity:

1. Research: With the help of the internet and suggested books, each group should start the project by researching the formulas and concepts needed to calculate the area of a pyramid.

2. Model Planning: Next, the group should decide what type of pyramid they will build (triangular, quadrangular, etc.) and choose an appropriate scale for the model. The group should make the necessary calculations to decide on the dimensions of the pyramid in the chosen scale.

3. Model Construction: With all dimensions determined, the group should cut the cardboard or cardstock into the appropriate shapes and sizes and assemble the pyramid.

4. Area Calculations: After assembling the model, the group should calculate the lateral surface area and the total surface area of the pyramid in the chosen scale. It is important to note that the area should be calculated as if the pyramid were real-sized, not in the model's scale.

5. Additional Research and Report: Finally, the group should research real-world applications of pyramid area calculations on the internet and write a report of at least 4 pages about the project, following the suggested structure:

   • Introduction: Describe the project's objective, the relevance of pyramid area calculations in the real world, and the chosen scale.
   • Development: Explain the concepts and formulas used to calculate the area of a pyramid, the model construction, and the methodology used for the area calculations.
   • Conclusion: Reflect on the construction and calculation process, and the lessons learned during the project.
   • Bibliography: List all resources used for the project's development.

By completing all these steps, the group will have carried out a comprehensive project on pyramid areas, from theoretical understanding, through the construction of a model, to the practical application of these concepts through area calculations and their correspondence with real-world situations.