Contextualization

Pyramids are figures frequently addressed in the study of Geometry, a subdiscipline of Mathematics that deals with the nature and properties of space. Unlike a prism, a pyramid has a base (which can be any polygon) and all other faces meet at a single point called the apex. In this project, we will focus on understanding and calculating the area of these geometric figures.

The surface area of a pyramid is calculated by adding the area of the base with the area of the lateral faces. Understanding how to calculate these areas is fundamental, as this knowledge can be applied in various situations, from solving practical problems to understanding more advanced concepts in mathematics and sciences.

In this scenario, the ability to calculate the area of a pyramid has practical applications that go beyond the academic world. Architects, engineers, designers, and many other professionals use this skill in their daily lives. Have you ever wondered why the pyramids of Egypt have a certain inclination to their faces? Well, that answer involves calculating the surface area of these monumental figures.

Introduction

In our study, we will use the following formula to calculate the surface area of a pyramid. If the base of the pyramid is a polygon with perimeter p and each lateral face is a triangle with height l (called the apothem), then the surface area A is given by:

A = 1/2 * p * l + A_base

Where A_base is the area of the base polygon.

Thus, the key to calculating this area is to understand how to calculate the area of a triangle and a polygon, skills that you should have acquired in previous years.

We recommend that you understand the theory behind the calculations well before attempting to apply them in a practical situation. For this, we suggest the following reliable resources:

• Khan Academy: Pyramid Area
• Math Dad: Pyramid Area
• World Education: Pyramids

Practical Activity

Activity Title: Modeling and Calculation of Pyramid Areas

Project Objective:

This project aims to develop the ability to calculate the area of a pyramid, as well as enhance teamwork skills, problem-solving, and practical application of theoretical concepts in a challenging and engaging context.

Detailed Project Description:

In groups of 3 to 5 students, you will be challenged to design and build a three-dimensional model of a pyramid using the suggested materials. Then, each group will calculate the surface area of the constructed model, with the help of the learned formulas. Finally, each group will prepare a report explaining the entire construction process and calculations carried out in the project.

Required Materials:

• Cardboard or cardstock;
• Ruler;
• Pencil;
• Pen;
• Scissors;
• Glue;
• Calculator.

Detailed Step-by-Step for Activity Execution:

1. Project Drawing: On cardboard or cardstock, each group will draw the flat projection of the pyramid model they intend to build. Remember to use the ruler to make precise drawings and correctly calculate the dimensions of all sides of the pyramid.

2. Construction of the 3D Model: Using scissors and glue, the groups will cut the flat projection and assemble it, forming the pyramid models.

3. Calculations: With the model ready, each group will carefully measure the sides of the pyramid base and the heights of the triangles that make up the lateral faces to calculate the total surface area of the pyramid using geometry formulas.

4. Report Writing: The final step is to write a detailed report contextualizing the relevance and application of the project, the theory studied, the detailed description of the activity, the calculation methodology used, the results obtained, conclusions, and bibliography.

Project Deliverables:

At the end of the project, each group must deliver:

1. 3D Pyramid Model: The three-dimensional model of the pyramid built by the group.

2. Detailed Report: The document should include the following topics:

   1. Introduction: Description of the relevance of calculating the area of a pyramid and where it can be applied.

   2. Development: Discussion on the theory behind the construction of the pyramid and the area calculation, describe the methodology used and explain the activity carried out by the group in detail, present the area calculation results obtained by the group.

   3. Conclusions: Discussion on the learnings obtained and the conclusions drawn about the project.

   4. Bibliography: Sources that the group used to learn about the project and to help in the construction of the pyramid and area calculation.