Area: Composite Figures | Traditional Summary
Contextualization
The calculation of areas of composite figures is a fundamental skill in Mathematics, especially for 7th-grade students in elementary school. Composite figures are formed by combining two or more simple geometric figures, such as triangles, rectangles, circles, among others. Understanding how to decompose these figures into smaller parts and calculate their individual areas is essential to solving practical everyday problems, such as determining the total area of a plot of land or an irregular room in a house.
This knowledge is widely used in various professions, such as architecture and engineering, where it is necessary to calculate the area of different parts of a construction to plan the amount of materials needed and the total cost of a project. Moreover, knowing how to calculate the area of composite figures can be useful in everyday situations, such as measuring spaces to install floors, carpets, or furniture. Therefore, mastering this skill not only improves mathematical understanding but also prepares students for practical applications in their future lives.
Introduction to the Concept of Composite Figures
Composite figures are formed by the combination of two or more simple geometric figures, such as triangles, rectangles, circles, among others. Decomposing these figures into smaller parts makes it easier to calculate the total area because it allows the application of known formulas for simpler figures.
The first step in calculating the area of a composite figure is to identify and separate the simple figures that compose it. This can be done by drawing auxiliary lines or considering the natural edges of the figures. Once identified, the areas of the simple figures are calculated individually and then summed to obtain the total area of the composite figure.
This approach is essential for solving practical problems, such as calculating the area of land with an irregular shape or the area of a house's floor plan. Decomposing into simple figures makes the calculation process more organized and less prone to errors.
-
Composite figures are combinations of simple geometric figures.
-
Decomposing into simple figures facilitates the calculation of the total area.
-
The areas of the simple figures are summed to obtain the total area.
Practical Example 1 - Area of a House
To illustrate the calculation of the area of composite figures, consider a simplified floor plan of a house composed of several rectangular and triangular rooms. The decomposition of this floor plan into simple figures allows the calculation of the total area of each room and consequently of the entire house.
For example, a rectangular room can be divided into two right triangles, and their areas can be calculated separately. Another room may simply be a rectangle, whose area is calculated by multiplying the base by the height. After calculating the area of each room individually, we sum all the areas to get the total area of the floor plan.
This type of calculation is common in architectural projects, where it is necessary to determine the total area to plan the amount of building materials and the cost of the work. The practice of decomposing complex figures into simple figures is a valuable skill for mathematics students and future professionals in the field.
-
Decomposition of the floor plan into simple figures.
-
Separate calculation of the area of each room.
-
Sum of the areas to obtain the total area of the house.
Practical Example 2 - Area of a Garden
Imagine a garden that has a rectangular part and a triangular part. To calculate the total area of the garden, it is necessary to decompose the space into its basic figures and calculate the area of each one separately.
The rectangular part of the garden can be easily calculated by multiplying the base by the height. Meanwhile, the triangular part requires the application of the area formula for a triangle, which is the base times the height divided by two. After obtaining the areas of both parts, we sum the values to determine the total area of the garden.
This method is useful not only for gardens but also for any external or internal space that has a combination of geometric shapes. It is a practical skill that can be applied in various contexts, from planning a park to organizing furniture in a space.
-
Decomposition of the garden into rectangular and triangular parts.
-
Separate calculation of the areas of the parts.
-
Sum of the areas to obtain the total area of the garden.
Tips and Strategies for Decomposition of Composite Figures
To facilitate the decomposition of composite figures, it is important to follow some tips and strategies. First, look for lines of symmetry or natural edges that can help divide the figure into smaller parts. These lines can be drawn manually or visualized mentally.
Another strategy is to identify overlapping or nested figures within the composite figure. By separating these figures, the calculation of their areas becomes simpler and more organized. Moreover, it is useful to review and memorize the basic area formulas, such as those for triangles, rectangles, and circles, to apply them quickly during the decomposition process.
Finally, practice decomposing composite figures in different contexts and problems. Constant practice helps develop the skill to quickly identify the simple figures that compose a complex figure and perform the necessary calculations efficiently.
-
Look for lines of symmetry or natural edges.
-
Identify overlapping or nested figures.
-
Review and memorize basic area formulas.
-
Practice decomposition in different contexts.
To Remember
-
Area: Measure of the surface of a geometric figure.
-
Composite Figures: Formed by the combination of simple geometric figures.
-
Decomposition: Process of dividing a composite figure into simple figures.
-
Triangle: Geometric figure with three sides and three vertices.
-
Rectangle: Geometric figure with four sides and right angles.
-
Area Formulas: Mathematical expressions used to calculate the area of simple geometric figures.
Conclusion
During the class, we discussed the importance of understanding and calculating the area of composite figures formed by the combination of simple geometric figures like triangles and rectangles. We learned that the decomposition of these figures into smaller parts facilitates the calculation of the total area, using known formulas for simpler figures. Additionally, we saw practical examples, such as the floor plan of a house and a garden, to illustrate the application of these concepts in real situations.
The ability to calculate composite areas is essential not only in mathematics but also in various professions such as architecture and engineering, where it is necessary to plan the amount of materials and the total cost of a project. Knowing how to decompose complex figures into simple ones makes the calculation process more organized and accurate, and it is a practical skill for everyday life.
Finally, we provided tips and strategies to facilitate the decomposition of composite figures, such as looking for lines of symmetry, identifying overlapping figures, and reviewing basic area formulas. These strategies, together with constant practice, help develop the skill of efficiently and effectively solving problems involving composite areas.
Study Tips
-
Constantly review the basic area formulas, such as those for triangles, rectangles, and circles, to become familiar with them and apply them quickly in different contexts.
-
Practice decomposing composite figures at home by drawing different geometric shapes and calculating their individual areas to sum them later.
-
Seek to solve practical everyday problems that involve the calculation of composite areas, such as measuring spaces in your home or garden, to apply the knowledge acquired in a practical way.
