Geometry (polygon)

Polygons

A polygon is a closed figure where the sides are all line segments. Each side must intersect exactly two others sides but only at their endpoints. The sides must be noncollinear and have a common endpoint.

A polygon is usually named after how many sides it has, a polygon with n-sides is called a n-gon. E.g. the building which houses United States Department of Defense is called pentagon since it has 5 sides.

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Types of Polygons

Regular or Irregular

A regular polygon has all angles equal and all sides equal, otherwise it is irregular

Regular		Irregular

Concave or Convex

A convex polygon has no angles pointing inwards. More precisely, no internal angle can be more than 180°.

If any internal angle is greater than 180° then the polygon is concave. (Think: concave has a "cave" in it)

Convex		Concave

Simple or Complex

A simple polygon has only one boundary, and it doesn't cross over itself. A complex polygon intersects itself! Many rules about polygons don't work when it is complex.

Simple Polygon (this one's a Pentagon)		Complex Polygon (also a Pentagon)

Interior angles of polygon

An Interior Angle is an angle inside a shape

Sum of Interior Angles = (n-2) × 180°

If a polygon is regular (i.e. all of its sides are of equal length and all of its angles are of equal size), then the following rule is true:

Each Interior Angle of a Regular Polygon = (n-2) × 180° / n

Where 'n' refers to the number of sides.

Exterior angles of polygons

The exterior angle of a polygon is an angle between any side of a shape and a line extended from the next side.

When you add the exterior angle of a polygon and its corresponding and exterior angle on a straight line, it amounts to 180°.

As a rule;

The sum of the exterior angles of a simple polygon add up to 360°

This is referred to as a 'full revolution'.

The exterior angle of a polygon and its corresponding interior angle always add up to 180°. This is due to the fact that the two corresponding angles create a straight line:

Exterior angle of a regular polygon = 360° ÷ (number of sides)

This is due to the fact that the exterior angles of a regular polygon always add up to 360°.

Tables of polygon and their angles