Polygon Angles | Traditional Summary
Contextualization
Angles are a fundamental part of geometry and have a significant presence in various areas of our daily lives, such as architecture, object design, and vehicle navigation. They are essential for creating stable and aesthetically pleasing structures. Today, our focus will be on the angles of polygons, which are geometric figures formed by line segments that meet at points called vertices. Polygons are closed shapes that include triangles, squares, pentagons, among others, and each of them has specific mathematical properties related to their internal and external angles.
Understanding how to calculate the internal and external angles of polygons is crucial not only for solving mathematical problems but also for practical applications in various professions. For example, architects and engineers need to calculate these angles to ensure the safety and functionality of their buildings. Additionally, programmers and graphic designers use these concepts to create three-dimensional graphics in video games and animations. By learning about the angles of polygons, you will be expanding your mathematical knowledge and your ability to apply these concepts in real-life situations.
Definition and Properties of Polygons
A polygon is a closed geometric figure formed by line segments that meet at points called vertices. These segments are called the sides of the polygon. Polygons can be classified in various ways, but one of the most common classifications is between regular and irregular polygons. A regular polygon has all sides and internal angles equal, while an irregular polygon does not have this uniformity.
Polygons can also be classified by the number of sides. For example, a triangle has three sides, a square or quadrilateral has four sides, a pentagon has five sides, and so on. Each type of polygon has specific properties related to its internal and external angles, which are essential for solving geometric problems.
Understanding the properties of polygons is essential for calculating angles and solving geometric problems. Knowing how to identify and classify polygons helps us apply the correct formulas to find their angular measures and other properties. Additionally, these properties are used in various practical applications, such as in the construction of buildings and the design of objects.
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A polygon is a closed geometric figure formed by line segments.
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Polygons can be regular (with equal sides and angles) or irregular.
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The classification of polygons can be done by the number of sides they have.
Sum of Internal Angles of Polygons
The sum of the internal angles of a polygon can be calculated using the formula (n-2) * 180°, where n is the number of sides of the polygon. This formula is derived from the fact that any polygon can be divided into triangles, and the sum of the internal angles of a triangle is always 180°. For example, a quadrilateral can be divided into two triangles, so the sum of the internal angles of a quadrilateral is 2 * 180° = 360°.
Applying this formula to different polygons allows us to calculate the sum of the internal angles quickly and efficiently. For example, for a pentagon (5 sides), the sum of the internal angles would be (5-2) * 180° = 3 * 180° = 540°. For a hexagon (6 sides), the sum would be (6-2) * 180° = 4 * 180° = 720°.
Understanding this formula is crucial for solving geometric problems and for the construction of mosaics and tiling, where angle precision is essential. Correctly applying this formula ensures that constructions are stable and aesthetically pleasing.
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The sum of the internal angles of a polygon is given by the formula (n-2) * 180°.
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This formula is based on the fact that any polygon can be divided into triangles.
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Applying the formula is essential for solving geometric problems and constructing mosaics and tiling.
Internal Angles of Regular Polygons
In regular polygons, all internal angles are equal. To find the measure of each internal angle, we can use the formula for the sum of the internal angles and divide by the number of sides of the polygon. For example, for a regular pentagon, we know that the sum of the internal angles is 540° (as calculated earlier). Since the pentagon has 5 sides, each internal angle is 540° / 5 = 108°.
This property is especially useful in constructing regular geometric figures and in solving problems involving angles. For example, in a regular hexagon (6 sides), the sum of the internal angles is 720°, and each internal angle is 720° / 6 = 120°. Knowing this property allows us to quickly calculate the angles of any regular polygon.
Additionally, this property is widely used in design and architecture. Structures that use regular polygons, such as tiles and panels, rely on the precision of internal angles to ensure that all pieces fit perfectly.
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In regular polygons, all internal angles are equal.
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The measure of each internal angle can be found by dividing the sum of the internal angles by the number of sides.
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This property is useful in constructing regular geometric figures and in design and architecture.
Relationship between Internal and External Angles
The external angles of a polygon are the angles formed between one side of the polygon and the extension of the adjacent side. An important property is that the sum of the external angles of any polygon is always 360°, regardless of the number of sides of the polygon. This can be visualized by considering that by traversing the entire perimeter of the polygon, you complete a full turn of 360°.
The relationship between internal and external angles is that each pair of internal and external angles at a vertex of a regular polygon sums to 180°. For example, in a regular hexagon, where each internal angle is 120°, the corresponding external angle would be 180° - 120° = 60°. This relationship is useful for solving geometric problems and understanding the properties of polygons.
This property is applied in various areas, such as in the construction of complex geometric shapes and the analysis of structures. Understanding the relationship between internal and external angles helps ensure accuracy and functionality in engineering and design projects.
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The sum of the external angles of any polygon is always 360°.
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Each pair of internal and external angles at a vertex of a regular polygon sums to 180°.
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This relationship is essential for solving geometric problems and applying in engineering and design projects.
To Remember
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Polygon: Closed geometric figure formed by line segments.
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Regular Polygon: Polygon with all sides and internal angles equal.
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Irregular Polygon: Polygon that does not have equal sides and internal angles.
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Internal Angles: Angles formed inside a polygon.
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External Angles: Angles formed between one side of the polygon and the extension of the adjacent side.
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Sum of Internal Angles: Total sum of internal angles of a polygon, calculated using the formula (n-2) * 180°.
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Sum of External Angles: Total sum of external angles of a polygon, always equal to 360°.
Conclusion
In this lesson, we learned about the definition and properties of polygons, focusing on their classifications and fundamental characteristics. We understood how to calculate the sum of the internal angles of polygons using the formula (n-2) * 180°, facilitating the resolution of geometric problems and practical applications. Additionally, we addressed the relationship between internal and external angles, highlighting that the sum of the external angles of any polygon is always 360°.
The importance of this knowledge extends to various areas, such as architecture, engineering, and graphic design, where the precision of angles is crucial to ensure the stability and aesthetics of structures. Understanding these properties helps apply mathematical concepts in real-life situations, such as in the construction of buildings and the creation of three-dimensional graphics.
We encourage students to explore more about the subject, as mastering these concepts enhances their mathematical skills and their ability to solve practical problems. Deeply understanding the properties of angles in polygons is essential for any student who wishes to excel in areas involving geometry and design.
Study Tips
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Review the concepts and formulas presented in class, practicing with different types of polygons to solidify your understanding.
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Use additional materials, such as geometry books and online resources, to deepen your knowledge of internal and external angles of polygons.
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Perform practical exercises, such as constructing mosaics or tiling, to apply the concepts learned and observe the practical application of angles in polygons.
