Interior Angles of a Polygon

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).

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

Reasoning

Triangle (3 sides): 180, Quadrilateral (4 sides): 180*2, Pentagon (5 sides): 180*3, Hexagon (6 sides): 180*4, n-gon (n sides): 180(n - 2)

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

Find the sum of the measures of the interior angles of a heptagon.

A heptagon has 7 sides.

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

Example 2

Find the sum of the measures of an interior angle of a regular octagon.

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.