Angles of Polygons

In this lesson, 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.

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 = 540º

The sum of interior angles in a pentagon is 540º.

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° 

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°.

Worksheet using the Formula for the Sum of Interior Angles

The following video shows 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

The following video shows how to find a missing angle using the sum of interior angles of a polygon.

The following videos use the sum of interior angles to write an equation and solve for the unknown.

Formula for the sum of exterior angles

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

 

Worksheet using the formula for the sum of exterior angles

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

The following video shows how to find the sum of the exterior angles and interior angles of a polygon and to 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.

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