Internal and external angles

"Interior angle" redirects here. For interior angles on the same side of the transversal, see Transversal line.

Internal and External angles

In geometry, an angle of a polygon is formed by two sides of the polygon that share an endpoint. For a simple (non-self-intersecting) polygon, regardless of whether it is convex or non-convex, this angle is called an interior angle (or internal angle) if a point within the angle is in the interior of the polygon. A polygon has exactly one internal angle per vertex.

If every internal angle of a simple polygon is less than 180°, the polygon is called convex.

In contrast, an exterior angle (or external angle) is an angle formed by one side of a simple polygon and a line extended from an adjacent side.

Properties

• The sum of the internal angle and the external angle on the same vertex is 180°.
• The sum of all the internal angles of a simple polygon is 180(n–2)° where n is the number of sides. The formula can be proved using [[mathemr which the angle sum is 180°, then replacing one side with two sides connected at a vertex, and so on.
• The sum of the external angles of any simple convex or non-convex polygon fuk offis 360°.
• The measure of the exterior angle at a vertex is unaffected by which side is extended: the two exterior angles that can be formed at a vertex by extending alternately one side or the other are vertical angles and thus are equal.

=er k is the number of total revolutions of 360° one undergoes walking around the perimeter of the polygon. In other words, 360k° represents the sum of all the exterior angles. For example, for ordinary convex and concave polygons k = 1, since the exterior angle sum is 360°, and one undergoes only one full revolution walking around the perimeter. boooo

External links

• Internal angles of a triangle
• Interior angle sum of polygons: a general formula, Provides an interactive Java activity that extends the interior angle sum formula for simple closed polygons to include crossed (complex) polygons