Polygons: Introduction | Active Summary

Objectives

1.  Understand what regular polygons are and be able to describe their properties, sides, and angles.

2.  Develop the ability to construct regular polygons using flowcharts or algorithms, knowing the measures of their sides.

3.  Apply the acquired knowledge about polygons in practical contexts, such as design and architecture, recognizing the importance of these geometric figures in our daily lives.

Contextualization

Did you know that polygons are everywhere in our world, from the architecture of iconic buildings to clothing patterns and even the packaging we use every day? For example, the iconic National Stadium of Beijing, known as the Bird's Nest, was inspired by geometric polygons. Mastering knowledge about polygons is not just a mathematical skill but also a crucial tool for many creative and technical professions.

Important Topics

Definition of Regular Polygons

Regular polygons are flat geometric figures formed by line segments called sides, which are connected by vertices. All sides and all internal angles of a regular polygon are congruent, meaning they have the same measure. This means that each vertex of a regular polygon is surrounded by the same number of sides.

• All regular polygons are convex, which means they do not have internal angles greater than 180 degrees, simplifying the understanding of their properties.

• The sum of the internal angles of a polygon with n sides is given by the formula (n-2) x 180 degrees, which is crucial for calculations and drawings in geometry.

• The measure of each internal angle of a regular polygon can be found by dividing the sum of the internal angles by the number of sides, facilitating the construction and visual verification of polygons.

Construction of Regular Polygons

The construction of regular polygons requires the correct application of measures of sides and angles, usually done using flowcharts or algorithms. This process is essential in various fields such as architecture and design, where precision in measurements and shapes is fundamental.

• Flowcharts and algorithms are visual and logical tools that help organize step by step the construction of polygons, facilitating the understanding and execution of the process.

• The measures of sides and angles can be calculated based on trigonometric knowledge or through specific formulas for each type of polygon, such as triangles, squares, and pentagons.

• Practice in constructing regular polygons not only reinforces mathematical understanding but also develops skills in precision and planning in practical tasks.

Practical Applications of Polygons

Regular polygons have practical applications in various areas, such as in architecture (space planning, facade design), in engineering (geometric structures, area calculations), and in graphic design (creating logos, layouts). Understanding their properties and how to construct them is fundamental for professionals in these fields.

• In architecture, the use of regular polygons can make a project more aesthetically pleasing and functional at the same time, as seen in the design of many modern stadiums and buildings.

• In engineering, knowledge about polygons helps in calculating loads and structures, where the regularity of shapes simplifies analyses.

• In graphic design, regular polygons are often used to create patterns and shapes that repeat, thanks to their symmetry and regularity, facilitating the creative process.

Key Terms

• Regular Polygon: A polygon whose sides have the same measure and whose internal angles are all equal.

• Vertex: The point where two or more sides of a polygon meet.

• Internal Angle: Each of the angles formed by two consecutive sides of a polygon.

To Reflect

• How do the symmetry and regularity of regular polygons influence their applications in design and architecture?

• In what ways can the understanding of geometric properties, such as the sum of internal angles, be applied in everyday situations or other areas of knowledge?

• What is the importance of using flowcharts and algorithms in the construction of regular polygons to ensure precision and efficiency in the final result?

Important Conclusions

• Today, we delved into the fascinating world of regular polygons, exploring their properties, construction, and practical applications.

• We discovered that regular polygons are more than geometric shapes; they are essential tools in areas such as architecture, engineering, and design, due to their symmetry and regularity.

• We learned to calculate the measure of internal angles, the total sum of internal angles, and how to construct polygons using flowcharts and algorithms—skills that have applications far beyond the academic world.

To Exercise Knowledge

1. Draw your polygonal world: Use graph paper to draw different regular polygons, such as triangles, squares, and hexagons, varying the size of their sides. 2. Geometric garden design: Imagine that you are a landscape designer and draw a small garden using only regular polygons. Calculate the areas that each polygon will occupy. 3. Modeling with sticks: Using popsicle sticks, create models of regular polygons and check their properties, such as the sum of internal angles.

Challenge

 Logo Designer Challenge: Create a logo using only regular polygons. Try to incorporate the symmetry and regularity of the polygons to convey a message or represent an idea. Present your logo and explain the choice of polygons and the symbolism behind them.

Study Tips

• Use geometric drawing apps to practice constructing regular polygons and visualizing their properties.

• Discuss with friends or family about the practical applications of polygons, such as in board games or in nature, reinforcing learning with everyday examples.

• Make visual summaries of the concepts learned, such as mind maps, to help memorize and understand the properties of polygons.