Introduction

Geometry is a subdiscipline of mathematics that deals with the study of the properties of shapes in space, including size, shape, position, direction, and spatial properties such as area and volume. In this project, we will focus on the study of the cylinder, a three-dimensional geometric figure formed by two congruent circles as bases and a curved lateral surface.

The area of a cylinder is composed of two parts: the lateral area and the area of the bases. The lateral area of a cylinder is the product of the circumference of the base (2 * pi * r) by the height (h) of the cylinder, and the area of the bases of a cylinder is the product of the area of a circle (pi * r²) by the number of bases (2). Therefore, the total area of a cylinder is the sum of the lateral area and the area of the bases.

The area of the cylinder is useful for understanding and calculating relevant concepts in various disciplines and contexts, ranging from mathematics, physics, engineering, architecture, and even design. Knowing how to calculate the area of a cylinder is, therefore, a valuable skill to solve real problems involving volume and capacity.

Contextualization

Cylinders are all around us, in food and hygiene product packaging, in machine and engine parts, in building and bridge structures, in furniture and electronic device design, and even in nature, in the form of tree trunks and basalt columns. Each application has specific requirements that depend on the cylinder's area to be met.

For example, an architect designing a cylindrical building needs to calculate the area of the building's surfaces to determine the amount of material needed. A packaging engineer needs to calculate the area of a soda can to know how much aluminum will be used in production. In nature, the surface area of a cylinder can help determine the age of a tree, as the trunk's diameter correlates with the number of growth rings.

Practical Activity

Multidisciplinary Project: "The Cylinder in Architecture and Engineering"

Project Objective

The objective of this project is to apply the mathematical concepts of the cylinder's area in a real context, exploring its applications in architecture and engineering. Students are challenged to design and build a physical model of a cylindrical building, calculate its areas, and justify the project's choices.

Detailed Project Description

Students, divided into groups of 3 to 5 people, must develop a cylindrical building, using knowledge acquired in mathematics (area, volume, and proportion calculations), physics (force structure), and arts (visual conception and project aesthetics). The project includes three stages: research, design, and construction of the building model, and calculation of the building's area.

This activity should take two to three weeks to complete, with students working a few hours each day.

Required Materials

• Cardboard, cardstock, or graph paper
• Ruler, compass, scissors, tape, and glue
• Pencils, colored pencils, pens, markers
• Calculator
• Internet access for research
• Books and educational materials as references

Detailed Step-by-Step for Activity Completion

1. Research (4 to 5 days): Students should research famous cylindrical buildings around the world (such as the Torre Agbar in Barcelona or the Schwäbisch Gmünd Water Tower in Germany). They should understand the architectural, aesthetic, and structural reasons for choosing the cylindrical shape.

2. Design (3 to 5 days): Each group must sketch a design of a cylindrical building. Students can choose to mimic an existing building or create an original design. The design should include drawings of the floor plan, front elevation, and cross-section.

3. Model Construction (5 to 7 days): Students must build a physical model of their building using the available materials. They should measure the radius and height of the building and use these measurements to calculate the lateral and total area of the building's surface.

4. Area Calculation (2 days): Based on the building's measurements, students must calculate the lateral and total area of the building and verify if the measurements match what was initially designed.

5. Report Development (4 to 5 days): Students must work together to prepare a detailed report on the project. The report should include: an introduction on the theme and building choice, a discussion on the project's design, a detailed description of the area calculations, and a conclusion summarizing the results and lessons learned.

Project Deliverables

Students must deliver:

1. Physical model of the cylindrical building.
2. Architectural project drawing (sketches and floor plans).
3. Calculations of the lateral and total areas of the building.
4. Detailed written document, containing:
   • Introduction: Description of the theme, building choice, and project objective.
   • Development: Detailing the research and design process of the building, description of the model construction, discussion of area calculations and their results.
   • Conclusions: Reflections on encountered problems and applied solutions, learnings and lessons obtained.
   • Bibliography: List of all sources consulted during the project.

These deliverables should demonstrate that students have acquired technical skills (cylinder area calculation) and socio-emotional skills (teamwork, time management, problem-solving).