# Angles of Polygons

In these lessons, we will learn
• how to calculate the sum of interior angles of a polygon using the sum of angles in a triangle
• the formula for the sum of interior angles in a polygon
• how to solve problems using the sum of interior angles
• the formula for the sum of exterior angles in a polygon
• how to solve problems using the sum of exterior angles

All the polygons in this lesson are assumed to be convex polygons.

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The following table gives the formula for the sum of the interior angles of a polygon and the sum of exterior angles of a polygon. Scroll down the page if you need more examples and explanation.

### Sum of Interior Angles of a Polygon

We first start with a triangle (which is a polygon with the fewest number of sides). We know that

The sum of interior angles in a triangle is 180º.

This is also called the Triangle Sum Theorem. Click here if you need a proof of the Triangle Sum Theorem.

Next, we can figure out the sum of interior angles of any polygon by dividing the polygon into triangles. We can separate a polygon into triangles by drawing all the diagonals that can be drawn from one single vertex.

In the quadrilateral shown below, we can draw only one diagonal from vertex A to vertex B. So, a quadrilateral can be separated into two triangles.

The sum of angles in a triangle is 180º. Since a quadrilateral is made up of two triangles the sum of its angles would be 180º × 2 = 360º

The sum of interior angles in a quadrilateral is 360º

A pentagon (five-sided polygon) can be divided into three triangles. The sum of its angles will be 180º × 3 = 560º

The sum of interior angles in a pentagon is 560º.

A hexagon (six-sided polygon) can be divided into four triangles. The sum of its angles will be 180º × 4 = 720º

The sum of interior angles in a hexagon is 720º.

### Formula for the sum of interior angles

We can see from the above examples that the number of triangles in a polygon is always two less than the number of sides of the polygon. We can then generalize the results for a n-sided polygon to get a formula to find the sum of the interior angles of any polygon.

If n is the number of sides of the polygon then,

Sum of interior angles = (n - 2) × 180°

For a regular polygon,

Each interior angle (of a regular polygon) =

Example:

Find the sum of the interior angles of a heptagon (7-sided)

Solution:

Step 1: Write down the formula (n - 2) × 180°

Step 2: Plug in the values           (7 - 2) × 180° = 5 × 180° = 900°

Answer: The sum of the interior angles of a heptagon (7-sided) is 900°.

Example:

Find the interior angle of a regular octagon.

Solution:

Step 1: Write down the formula

Step 2: Plug in the values

Answer: Each interior angle of an otagon (8-sided) is 135°.

Worksheet using the Formula for the Sum of Interior Angles

How to find the sum of the interior angles of any polygon using triangles and then derive the generalized formula?

### Problems using the sum of interior angles

How to find a missing angle using the sum of interior angles of a polygon? How to use the sum of interior angles to write an equation and solve for the unknown?
Write an equation and solve for the unknown. Substitute your answer into each expression to determine the measure of the angle. Give reasons for your answers.

### Formula for the sum of exterior angles

The sum of exterior angles of any polygon is 360º.

The exterior angle of a regular polygon is

Worksheet using the formula for the sum of exterior angles

Worksheet using the formula for the sum of interior and exterior angles

How to find the sum of the exterior angles and interior angles of a polygon?
Every convex polygon has interior and exterior angles. The interior angles are inside the polygon formed by the sides. The exterior angles form a linear pair with the interior angles.
Example:
Determine the measure of each exterior and interior angle of a regular polygon.

### Problems using the sum of exterior angles

The following video shows a problem involving the sum of exterior angles of a polygon.
Example:
A regular polygon has an exterior angle that measures 40°. How many sides does the polygon have?

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