# Interior Angles of a Polygon

How to solve the interior angles of a polygon problems: formula, examples, and their solutions.

## Formula

The sum of the measures of the interior angles of an *n*-gon is 180(*n* - 2).

## Reasoning

To understand this formula easily,

let's make an inductive reasoning

and find a pattern.

Triangle (3 sides):

(sum) = 180⋅1 = 180⋅(3 - 2).

Interior angles of a triangle

Quadrilateral (4 sides):

There are 2 triangles in a quadrilateral.

So (sum) = 180⋅2 = 180⋅(4 - 2).

Interior angles of a quadrilateral

Pentagon (5 sides):

There are 3 triangles in a pentagon.

So (sum) = 180⋅3 = 180⋅(5 - 2).

Hexagon (6 sides):

There are 4 triangles in a hexagon.

So (sum) = 180⋅4 = 180⋅(6 - 2).

*n*-gon (*n* sides):

So (sum) = 180(*n* - 2).

## Example 1

A heptagon has 7 sides.

So (sum) = 180(7 - 2).

## Example 2

An octagon has 8 sides.

So (sum) = 180(8 - 2) = 1080.

A regular octagon has 8 congruent sides

and 8 congruent interior angles.

So the measure of an interior angle

of a regular octagon is 1080/8 = 135.