Exploring Angles of Polygons: From Theory to Practice

Objectives

1. Calculate measures of internal angles of regular polygons.

2. Establish relationships between internal and external angles of polygons.

3. Develop logical reasoning skills and solve mathematical problems.

4. Promote the practical application of mathematical concepts in everyday situations and the job market.

Contextualization

The angles of polygons are present in various situations of our daily lives. When we observe architecture, for example, we notice the importance of accurately calculating angles for the construction of buildings and bridges. Designers use polygons and their angles to create aesthetic patterns in clothing, jewelry, and even in application interfaces. Understanding the angles of polygons is not just a theoretical issue but a practical skill that can be applied in many professional areas.

Relevance of the Theme

Understanding the angles of polygons is crucial in the current context, as it allows practical application in various professions such as architecture, engineering, graphic design, fashion, and robotics. Precision in calculating angles is fundamental to ensuring the functionality and aesthetics of projects, making it a valuable skill in the job market.

Internal Angles of Polygons

The internal angles of a polygon are the angles formed between two adjacent sides within the polygon. The sum of the internal angles of a polygon depends on the number of sides and can be calculated using the formula (n-2) * 180°, where n is the number of sides.

• The sum of the internal angles of a triangle is always 180°.

• For a square, the sum of the internal angles is 360°.

• In a polygon with n sides, the sum of the internal angles is (n-2) * 180°.

External Angles of Polygons

The external angles of a polygon are the angles formed between one side of the polygon and the extension of the adjacent side. The sum of the external angles of any polygon is always 360°, regardless of the number of sides.

• The sum of the external angles of a triangle is 360°.

• For any polygon, the sum of the external angles is always 360°.

• External angles help determine the shape and symmetry of regular polygons.

Regular Polygons

Regular polygons are those that have all sides and angles equal. In regular polygons, the internal and external angles are uniform, making it easier to calculate and construct symmetric models.

• A square is an example of a regular polygon with four equal sides and internal angles of 90°.

• In a regular hexagon, each internal angle measures 120°.

• Regular polygons are often used in design and architecture due to their symmetry.

Practical Applications

• In architecture, accurately calculating internal and external angles is essential for constructing stable and aesthetically pleasing buildings.
• In graphic design, regular polygons are used to create logos and symmetric patterns.
• In robotics, programming robot movements and trajectories depends on accurately calculating angles to ensure smooth and precise movements.

Key Terms

• Internal Angle: Angle formed between two adjacent sides of a polygon within the figure.

• External Angle: Angle formed between one side of the polygon and the extension of the adjacent side.

• Regular Polygon: Polygon that has all sides and angles equal, exhibiting uniform symmetry.

Questions

• How can precision in calculating internal and external angles affect the stability of an architectural structure?

• In what ways does the symmetry of regular polygons influence graphic design and the aesthetics of logos?

• What challenges are encountered when programming robots to perform smooth and precise movements, considering the angles of their trajectories?

Conclusion

To Reflect

Understanding the angles of polygons is more than just a theoretical issue. It is an essential skill that applies to various professional areas such as architecture, graphic design, robotics, and fashion. During this lesson, we explored how to calculate the internal and external angles of regular polygons and discussed the relevance of this skill in the job market. The practice of constructing and measuring polygons helped us solidify this knowledge, showing how mathematics can be applied in real and practical situations. By mastering these concepts, we are better prepared to face challenges and create innovative solutions in various fields.

Mini Challenge - Drawing Polygons in Everyday Life

The challenge consists of identifying and drawing polygons present in everyday life, applying knowledge about internal and external angles.

• Choose an object from your daily life that contains polygons (for example, a window, a tile, a logo, etc.).
• Draw the chosen polygon on a sheet of paper, detailing the sides and vertices.
• Use a protractor to measure and record the internal angles of the polygon.
• Calculate the sum of the internal angles and check if it corresponds to the formula (n-2) * 180°, where n is the number of sides.
• Determine and record the external angles of the polygon, verifying if the sum of the external angles is 360°.
• Write a brief reflection on how the precision in measuring angles influences the functionality and aesthetics of the chosen object.