Calculating the sum of polygons' interior and exterior angles

In this note, you will learn how to calculate the sum of the interior and exterior angles of polygons.

The sum of the interior angles of a polygon:
The formula for the sum of the interior angles of a polygon can be derived by dividing it into triangles.
Diagonal lines that do not cross each other can be drawn inside the polygon.
For example, a hexagon with six sides can be divided into four triangles, as shown in the figure below:

Remember, the sum of the interior angles of a triangle is 180°.
There are four triangles in the hexagon, therefore the sum of the interior angles of a hexagon is 4 × 180° = 720°.
A polygon will always be divided into the number of triangles that are two fewer than the number of sides.
A pentagon, which has five sides, can be divided into three triangles, and the sum of the interior angles is 3 × 180° = 540°.

In general:

The sum of the interior angles = (

n

— 2) × 180°
Where:

n

is the number of sides or angles of the polygon.

Example 1:
Calculate the sum of the interior angles of a polygon with twelve sides.

Solution:
Sum of interior angles = ( $n$ — 2) × 180°
                                 = (12 — 2) × 180º
                                 = 10 × 180º
                                 = 1 800°

Example 2:
Calculate the sum of the interior angles of a heptagon.

Solution:
A heptagon has seven sides.
Sum of interior angles = ( $n$ — 2) × 180°
                                 = (7 — 2) × 180º
                                 = 5 × 180º
                                 = 900°
The sum of the exterior angles of a polygon:
The sum of the exterior angles of a polygon is equal to one complete revolution(360º).
The sum of the exterior angles of a polygon does not depend on the number of sides or angles a polygon has.

In general:

The sum of the exterior angles = 360°

Example 3:
Determine the sum of the exterior angles of a polygon with fifteen sides.

Solution:
Sum of exterior angles = 360°

Example 4:
Determine the sum of the exterior angles of a nonagon.

Solution:
A nonagon has nine sides.
Sum of exterior angles = 360°

Activity:

1. Mention the number of triangles that can be formed by the following polygons.

a.                          b.
lozenge – Wiktionary
2. An octagon has eight sides.
   a. Draw an octagon and divide it into triangles by drawing diagonal lines.
   b. Determine the sum of the interior angles of an octagon.
   c. Determine the sum of the exterior angles of an octagon.

3. Copy and complete the table below:

Number of sides:	Number of triangles:	Sum of interior angles:
3
4
7
20

Solutions Click on the following link for Solutions:
Solutions Sum of interior and exterior angles of polygons.docx

References and Attributions

Note by H Alberto Gongora from Noun Project CC BY 3.0
Question mark by Vectorink from Wikimedia commons, CCO 1.0

This note is licensed under a Creative Commons Attribution-ShareAlike 4.0 International License

Get Started

Password contains a name or word of six or more letters with a capital, a special character (*&%) and numbers