Intermediate Geometry

Intermediate Geometry explores the properties and relationships of shapes, sizes, and theorems in two and three dimensions.

Basic Concepts

Polygons and Their Properties

Exploring Polygons

A polygon is a closed shape with straight sides. The number of sides determines the polygon's name (pentagon - 5, hexagon - 6, etc.).

Angle Sum Rule

The sum of the interior angles of a polygon is: \[ (n - 2) \times 180^\circ \] where \(n\) is the number of sides.

Regular vs. Irregular Polygons

  • Regular Polygon: All sides and angles are equal (e.g., regular hexagon).
  • Irregular Polygon: Sides and angles are not all equal.

Applications

Polygons are everywhere: stop signs (octagons), honeycombs (hexagons), and tiled floors often use regular polygons for symmetry and design.

Fun Fact

The more sides a regular polygon has, the closer it resembles a circle!

Examples

  • A pentagon has five sides and its interior angles sum to \(540^\circ\).

  • A soccer ball's surface is made up of hexagons and pentagons.

In a Nutshell

Polygons are multi-sided shapes with angle and side properties used in art, architecture, and nature.