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Polygons: Properties and Applications

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Polygons are two-dimensional shapes with straight, non-intersecting sides and a set number of vertices and angles. They range from simple triangles to complex n-gons, each with unique properties. This overview covers the basics of polygon classification by sides, regularity, and convexity, as well as methods for calculating interior and exterior angles and areas. Understanding these fundamental geometric figures is crucial for various mathematical and practical applications.

Exploring the Fundamentals of Polygons

A polygon is a two-dimensional geometric figure composed of a finite number of straight line segments connected to form a closed polygonal chain or circuit. These segments are referred to as edges or sides, and the points where two edges meet are the vertices. Polygons are primarily classified by the number of sides they possess, starting with the triangle at three sides. The interior angles of a polygon are the angles inside the polygon at each vertex, and the sum of these angles depends on the number of sides. Polygons must have non-intersecting sides; if the sides intersect, the figure is not considered a polygon. Additionally, polygons do not include figures with curved sides, as all sides must be straight.
Assorted geometric shapes including a translucent blue hexagon, yellow triangle, red square, orange pentagon, green rectangle, and a gradient purple polygon.

Categorizing Polygons by Regularity and Convexity

Polygons are differentiated by their regularity and convexity. A regular polygon has all sides and all interior angles equal, such as an equilateral triangle or a square. An irregular polygon, in contrast, has sides and angles of differing lengths and measures, such as a scalene triangle or a rectangle. Convex polygons have interior angles less than 180 degrees, and no line segment between any two points in the polygon will lie outside of it. A concave polygon has at least one interior angle greater than 180 degrees, and at least one line segment between points in the figure will pass outside of the polygon. These classifications help in understanding the properties and relationships of various polygon types.

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00

Polygon sides and vertices relationship

A polygon's sides are straight segments; vertices are points where sides meet.

01

Interior angles of a polygon

Sum of interior angles depends on number of sides; calculated by (n-2)*180 degrees for n-sided polygon.

02

Non-intersecting sides criterion

For a figure to be a polygon, its sides must not intersect each other.

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