Polygon angle sum theorem to have the formula for total sum of angles:

The polygon angle sum theorem is used to calculate total sum of all angles of a polygon. The polygon angle sum formula can be derived by dividing a polygon into number of triangles. We know that sum of all angles of a triangle is always equal to 180°. So angle sum of a polygon can be calculated by multiplying number of triangles with 180°. A quadrilateral can be divided in 2 triangles which gives the total angle of 360°. A pentagon can be divided into 3 triangles that makes polygon interior angle sum equal to 540°. If we move on like this, every polygon can be divided 1 more time as its previous one. This gives total angle sum of polygon equal to (n-2) * 180° where n is number of sides of a polygon.

Polygon interior angle sum theorem for basic Polygons:

From above discussion, it is clear that Polygon interior angle sum theorem yields sum of angles as (n-2) * 180°. In regular polygons, value of all the interior angles is same. Whereas for an irregular polygon, all angles are not same. Let's take an example of this. As you know, Octagon has 8 sides. Applying the angle formula, total angle for octagon is (8-2) * 180° = 6 * 180° = 1080°. For a regular octagon, each angle is equal to 1080/8 = 135°. For a irregular one, it will be different.

basic_in_polygons_shapes

Above figure shows some of the basic regular polygons up to 9 sides, number of sides, total interior angle & each interior angle. If you verify the formula results for all these polygons, you get correct results for interior angle sum of a polygon. 

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