Contextualization

The study of geometry, more precisely about the areas of plane figures, is one of the fundamental pillars of mathematics. Area is the measure of the 'size' of the surface of a figure or flat surface and is measured in square units. In this project, we will focus on one of the most basic concepts of geometry, a triangle - a closed three-dimensional geometric figure that has three sides and three vertices.

The formula for calculating the area of a triangle is one of the first ones we learn: Area = (base * height) / 2. But did you know that there are other formulas to calculate the area of a triangle based on different provided information? For example, if you know the three sides of a triangle, you can use the Heron's formula. If you have a right triangle, you can use the Pythagorean theorem to find the length of the sides.

Learning these important concepts will be reinforced through the creation of a project incorporating these ideas. But why is studying the calculation of triangle areas important? The ability to calculate the area of a triangle has countless applications in many fields, such as architecture, engineering, computer science, among others.

For example, in architecture, it is often necessary to calculate areas to design spaces. The area of the ground of a house or building is the product of the base by the height, just like the area of a triangle! In computer science, especially in the field of computer graphics, triangles are used to model three-dimensional surfaces. The ability to calculate the area of a triangle is essential for this process.

To delve deeper into how the area of triangles is calculated and its practical applications, you can check the following resources:

• Simple Mathematics - Triangle Area - This is an excellent resource to understand the basics of calculating the area of a triangle.
• Brazil School - Heron's Formula - This link has an excellent explanation of Heron's Formula to calculate the area of a triangle.
• Math Portal Channel - Triangle Area and Special Triangles - An informative video resource about triangle areas and special triangles.

Practical Activity

Activity Title: 'Triangles in Real Life: A Cross-Disciplinary Study'

Project Objective

This project aims to develop students' understanding of the triangle area, its different calculation formulas, and its practical applications. It also aims to develop important skills such as teamwork, time management, critical and creative thinking.

Detailed Project Description

Students will work in groups of 3 to 5, dedicating at least 12 hours each to carry out the project. Each group will be tasked with creating a model for a fictional city, integrating the calculation of the triangle area into various aspects of city planning and design. This project will cross mathematics and geography (urban planning) to help students understand the practical application of calculating triangle areas.

Required Materials

• Graph paper or cardboard paper.
• Pencils and erasers.
• Ruler and calculator.
• Colored pencils and/or markers.
• Computer with Internet access for research.

Detailed Step-by-Step

1. Research and Planning: Groups will start by researching city planning and how areas (such as building areas, parks, roads, etc.) are important in this process. Additionally, they should research how the calculation of triangle areas is used in reality.

2. City Design Sketch: Each group should draw a sketch of their city on a piece of paper, dividing the city into different areas, such as residential, commercial, industrial, among others. Each area should contain elements that are composed of triangular shapes, such as pyramids, zigzag roads, triangular soccer fields, etc.

3. Area Calculations: Using the learned concepts of triangle area, students will calculate the area of each triangle in their city. They will use different formulas for area calculation, depending on the available information (for example, if they only know the base and height, if they know all three sides, if it is a right triangle, etc.).

4. Geography Integration: Students will research how urban planning affects the geographical and environmental aspects of a region. They will explain the decisions they made when designing their city, such as the location of different areas and how they relate to the geography of their city.

5. Final Model Creation: Based on the sketch and calculations, students create the final model of their city, coloring and marking the different areas.

6. Final Report: Each group will prepare a detailed report explaining the theory behind the area calculation, the description of the process they followed to design their city, the methodology used to calculate the triangle areas, and the decisions made based on geographical aspects. The report should be divided into four main topics: Introduction, Development, Conclusions, and Bibliography used.

Project Deliverables

Each group must deliver the following items at the end of the project:

• The final model of the designed city.
• Detailed final report.

These deliveries will serve to assess students' understanding of the triangle area, their technical and socio-emotional skills, and their ability to apply theoretical knowledge to practical situations.