Sum of Interior Angle of Polygon Theorem

Statement: The interior angle theorem states the sum of the interior angles of a polygon with n vertices is S = (n – 2) × 180°.

Derivation:

To prove the interior angle theorem, we need the statement that the sum of the interior angles of a triangle is 180°.

Now, let us consider a polygon having “n” sides. A polygon having “n” sides forms “n” triangles. For example, look at the figure given below, where a polygon having 8 sides has formed 8 triangles.

As we know that the sum of the interior angles of a triangle is 180°, the sum of all the interior angles of “n” triangles will be n × 180°.

Sum of all the interior angles of “n” triangles = n × 180°

So, we can conclude that,

Sum of interior angles of the polygon + Sum of the angles at the point O = n × 180° ——— (1)

We know that the sum of the angles at the point O = 360° ——— (2)

By substituting equation (2) in equation (1), we get

Sum of interior angles of the polygon + 360°= n × 180°

So, the sum of interior angles of the polygon = n × 180° – 360°

= (n – 2) × 180° 

Thus, the sum of interior angles of the polygon = (n – 2) × 180°

The measure of each interior angle of a regular polygon = ((n – 2) × 180°/n) (Since the measure of each angle is the same for a regular polygon)

Hence proved.

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Polygon is defined as a two-dimensional geometric figure that has a finite number of line segments connected to form a closed shape. The line segments of a polygon are called edges or sides, and the point of intersection of two edges is called a vertex. The angle of a polygon is referred to as the space formed at the intersection point (vertex) of two adjacent sides.

A polygon is of two types: a regular polygon and an irregular polygon. A regular polygon is a polygon whose all sides and all interior angles are measured the same, whereas an irregular polygon is a polygon whose all sides and all interior angles do not measure the same. And we also have different types of polygons like triangles, quadrilaterals, pentagons, hexagons, etc, based on the number of sides of a polygon. Every polygon has interior angles and exterior angles, where an interior angle is the one that lies inside the polygon and the exterior angle is the one that lies outside the polygon.