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The Geometry of Polygons

From simple triangles to complex decagons, understanding multi-sided figures.

A Polygon is a closed 2D shape with straight sides. The word comes from Greek "poly" (many) and "gon" (angle). If a shape has curved lines or is not closed (leaving a gap), it is not a polygon. Polygons are the primary way we classify 2D space in geometry.

1. Classifying Polygons by Sides

We name polygons based on the number of sides (n) they possess. The most common include:

  • Triangle: 3 sides
  • Quadrilateral: 4 sides
  • Pentagon: 5 sides
  • Hexagon: 6 sides
  • Heptagon: 7 sides
  • Octagon: 8 sides
  • Nonagon: 9 sides
  • Decagon: 10 sides

2. Regular vs. Irregular Polygons

This distinction is crucial in geometry:

  • Regular Polygon: All sides are equal in length, and all interior angles are equal measure. Examples: Equilateral Triangle, Square.
  • Irregular Polygon: Sides and angles can be of any varying length or measure.

3. Interior Angle Formula

The sum of the interior angles of a polygon depends on the number of sides ($n$). The formula is:

Sum of Interior Angles = $(n - 2) \times 180^{\circ}$

For example, a Hexagon ($n=6$) has a sum of $(6-2) \times 180 = 720^{\circ}$.

Conclusion

Whether it's the honeycomb structure of a beehive (hexagons) or the stop sign on the street (octagons), Polygons are fundamental to structure and design in our world.