Area: Composite Figures | Active Summary

Objectives

1.  Master the calculation of areas of composite figures formed by triangles and rectangles.

2.  Solve practical everyday problems that involve area calculation, such as space or construction planning.

Contextualization

Did you know that calculating areas is an essential skill not only for mathematicians but also for professionals like architects, engineers, and even farmers? These composite figures that we will explore are the basis for designing complex structures and planning efficient spaces. Learning about them not only strengthens your mathematical skills but also opens doors to understanding how the world around us is designed and organized!

Important Topics

Triangles

Triangles are geometric figures that have three sides and three internal angles. In the context of composite figures, triangles can be combined with other shapes such as rectangles to form more complex structures. The area calculation of a triangle, essential for studying composite figures, is given by 1/2 of the base multiplied by the height.

• The calculation of a triangle's area is fundamental for understanding how they can be integrated into larger figures.

• The formula for calculating the area of a triangle, 1/2 * base * height, is an essential tool in projects that involve triangulations in engineering and architecture.

• The ability to decompose larger figures into triangles helps simplify area calculation problems, making them more accessible and understandable.

Rectangles

Rectangles are quadrilaterals with four right angles and equal opposite sides. They are the basis for understanding the composition of many composite figures due to their simplicity and regularity. The area calculation of a rectangle is done by multiplying the base by the height.

• Rectangles serve as building blocks for many more complex geometric figures.

• The area formula of a rectangle (base * height) is one of the first and most used formulas in geometry, applicable in countless practical situations.

• Understanding the property of rectangles to maintain the area unchanged under rotation or translation is crucial for design and engineering.

Composite Figures

Composite figures are formed by the combination of two or more simple geometric figures, such as triangles and rectangles. The ability to calculate the area of composite figures is essential for many practical applications, such as interior design, architecture, and urban planning. This calculation involves decomposing the composite figure into simpler figures, calculating their individual areas, and summing these areas.

• The ability to calculate the area of composite figures allows for optimization of spaces and resources, crucial in fields like urban planning and interior design.

• Decomposing composite figures into simpler figures facilitates the calculation and understanding of how different parts of a structure contribute to the total area.

• Problems involving composite figures develop logical reasoning and spatial visualization skills, valuable competencies in mathematics and sciences.

Key Terms

• Area of a Triangle: Measure of the space within the boundaries of a triangle, calculated as 1/2 * base * height.

• Area of a Rectangle: Space occupied by a rectangle, calculated as base * height.

• Composite Figures: Structures formed by the combination of two or more simple geometric figures, whose areas can be calculated by summing the areas of the component parts.

To Reflect

• How can the calculation of areas of composite figures impact the design of an urban park?

• Why is it important to understand the area of each component in a composite figure rather than just measuring the total perimeter?

• In what way does the study of areas in composite figures contribute to the development of problem-solving skills in other areas of knowledge?

Important Conclusions

• In this lesson, we explored the fascinating world of composite figures, learning how to calculate the area of structures formed by triangles and rectangles. We saw that this knowledge is not only mathematical but essential in various professions such as architecture, engineering, and even event planning.

• The ability to decompose complex figures into simple shapes and calculate their areas allows us to understand and optimize spaces, a crucial skill in many aspects of our daily lives and the professional world.

• We reinforced that the study of geometry is not just about formulas but also about practical application and solving real-world problems, making this knowledge incredibly valuable and relevant.

To Exercise Knowledge

1. Draw the layout of your room, including all furniture, and calculate the area occupied by each item. 2. Design a small amusement park in a rectangular space, using triangles to represent the areas of different attractions. 3. Create a map of your house, calculating the area of each room and comparing them to see how you can reorganize the space more efficiently.

Challenge

Junior Urban Planner Challenge: Use your skills to plan a small neighborhood on graph paper. Each house should consist of a rectangle (the house itself) and a triangle (the garden). Calculate the areas of each house and garden, ensuring that each house has an appropriate garden space.

Study Tips

• Practice drawing different composite figures on graph paper to better visualize the areas and how they add up.

• Try solving everyday area calculation problems, such as calculating the area of a package to wrap a gift or the amount of paint needed to paint an irregular wall.

• Use online resources like geometry simulators to visually explore how the areas of composite figures are calculated and how different configurations affect the final result.