Introduction

The study of geometric solids is a fundamental part of mathematics, and when we delve into the universe of cylinders, there are several practical applications that can be explored, from soda cans to integral components of a hydroelectric plant, for example. In this project, we will study two key concepts related to cylinders: volume and surface area.

The volume of a cylinder is the measure of the capacity that this solid has to contain some substance within it, whether it be a liquid, a gas, or even a solid. It is calculated using the formula Volume = π * radius² * height, where π is a mathematical constant approximately equal to 3.14159.

The surface area of a cylinder corresponds to the sum of the areas of its two bases with the lateral area. The latter is calculated through the formula Lateral Area = 2 * π * radius * height, while each base has an area of Base Area = π * radius². Therefore, the total area is given by Total Area = 2 * Base Area + Lateral Area.

Contextualization

Cylinders are present in our daily lives in various forms. A soda can, for example, is nothing more than a cylinder. Understanding the concept of volume and surface area allows us to understand things like: Why can a 350ml soda can seem larger or smaller depending on its shape? Or even how companies choose the size of their product packaging?

Furthermore, the understanding of these concepts is also linked to broader contexts. In engineering, for example, the volume and area of the cylinder are essential for the design of components such as hydraulic turbines and water reservoirs. In art, they are also used to calculate the necessary material for the construction of cylindrical sculptures.

Practical Activity

Activity Title: "Modeling and Measuring Cylinders"

Project Objectives

1. Develop the ability to calculate the surface area and volume of a cylinder.
2. Understand the practical application of these concepts.
3. Reinforce the ability to work in a team, meet deadlines, and communicate effectively.

Detailed Project Description

In this project, students will be divided into groups of 3 to 5 people. Each group must build models of cylinders with different dimensions, using recyclable materials (cardboard, old papers, adhesive tape, etc). After building the models, they must calculate the volume and surface area of each cylinder and compare the results between groups. Discussing the results and reflecting on the practical application of these concepts will be the main objectives of this activity.

Required Materials

• Cardboard or any type of recycled paper
• Adhesive tape
• Ruler or measuring tape
• Pens or pencils
• Calculator
• Scissors
• Decoration materials (optional)

Detailed Step-by-Step

1. Students will be divided into groups of 3 to 5 people.

2. Each group must build at least three models of cylinders with different dimensions. The cylinders should be built using recycled papers and adhesive tape.

3. After building the models, each group must measure the height and radius of each cylinder and calculate the volume and surface area using the learned formulas.

4. The groups must then compare the results obtained. They should discuss if the results are proportional to the dimensions of the cylinders and if they are in line with what they expected.

5. Each group must prepare a report where they explain the process of building the models, how they performed the calculations, and what the comparison between the results is.

6. Finally, the groups must reflect on the practical application of the concepts of volume and surface area of a cylinder. They can think about how these concepts are used in everyday life and cite examples of the use of these concepts. This content should be presented in the report.

Project Deliverables

As a result, students must present:

1. Models of cylinders, built with recycled paper;
2. Calculations of the volume and surface area of each cylinder;
3. Detailed project report, including:
   • Introduction: Contextualizing the theme, the relevance and application in the real world, and the objective of this project.
   • Development: Explaining the theory behind the volume and surface area of the cylinder, detailing the activity, the methodology used, and finally presenting and discussing the results obtained.
   • Conclusion: Summarizing the main points, explaining the learnings obtained, and the conclusions drawn from the project.
   • Bibliography: Indicating the sources they relied on to work on the project, such as books, web pages, videos, etc.