1. Classification by Number of Sides
One of the simplest ways to classify polygons is by counting how many sides they have. A polygon with \( n \) sides is called an \( n \)-gon. This naming helps us talk about polygons in a general way.
1.1. Common Names Based on Sides
| Number of Sides | Name of Polygon | Quick Example |
|---|---|---|
| 3 | Triangle | Roof truss shape |
| 4 | Quadrilateral | Rectangle, square |
| 5 | Pentagon | Some tile designs |
| 6 | Hexagon | Honeycomb cell |
| 7 | Heptagon | Decorative patterns |
| 8 | Octagon | Stop sign |
For any \( n \ge 3 \), a polygon with \( n \) sides can be called an \( n \)-gon. For example, a 10-sided polygon is a decagon.
2. Convex and Concave Polygons
Polygons can look outward or inward depending on how their angles are shaped. This gives us two important types: convex and concave polygons.
2.1. Convex Polygons
A polygon is convex if all its interior angles are less than \(180^\circ\), and the shape "bulges outward". If you draw a line through a convex polygon, it will never pass through the interior more than once.
Examples: square, equilateral triangle, regular pentagon.
2.2. Concave Polygons
A polygon is concave if at least one interior angle is greater than \(180^\circ\). Concave polygons look like they have a "caved-in" portion.
One simple test: If you can draw at least one diagonal that lies outside the polygon, then it is concave.
Examples: star-shaped polygons, arrow-shaped quadrilaterals.
3. Simple and Complex Polygons
Some polygons have neat, non-crossing sides, while others have sides that cross over each other. Based on this, we classify polygons into simple and complex types.
3.1. Simple Polygons
A polygon is simple if its sides intersect only at their endpoints (vertices). The boundary does not cross itself.
Examples: any triangle, rectangle, pentagon, hexagon that you commonly draw in geometry.
3.2. Complex (Self-Intersecting) Polygons
A polygon is complex if its sides intersect each other at points that are not endpoints. The shape looks twisted or overlapped.
Examples: star polygons or shapes made by crossing lines.
4. Regular and Irregular Polygons
This is another important way of classifying polygons based on the equality of sides and angles.
4.1. Regular Polygons
A polygon is regular if all its sides are equal and all its interior angles are equal. Such polygons have beautiful symmetry and look balanced from all sides.
Examples: equilateral triangle, square, regular hexagon.
4.2. Irregular Polygons
If a polygon does not have all equal sides and all equal angles, it is called an irregular polygon.
Examples: any typical quadrilateral or pentagon drawn freely by hand.
5. Quick Visual Summary (As Personal Notes)
Here is a neat way to remember the classification:
- By sides: triangle, quadrilateral, pentagon, hexagon, ...
- By shape: convex vs concave
- By boundary: simple vs complex
- By equality: regular vs irregular
These categories make it easier to talk about polygons, compare them, and use formulas for angles and diagonals in later topics.