Interior and Exterior Angles of Regular Polygons Word Problems

Concept

The angles that lie inside a shape, generally a polygon, are said to be interior angles.

An exterior angle of a polygon is the angle that is formed between any side of the polygon and a line extended from the next side. Every polygon has interior and exterior angles. The exterior is the term opposite to the interior which means outside. Therefore exterior angles can be found outside the polygon. The sum of the exterior angles of any polygon is equal to 360°. Any flat shape or figure is said to have interior or exterior angles only if it is a closed shape.

Rules

To find the sum of all interior angles in a regular polygon:
The sum of the interior angles of a polygon can be found by taking the number of sides (n) and subtracting 2. Then, multiply that number by 180.

Sum of interior angles = (n – 2) ∙ 180°

To find the measure of each interior angle in a regular polygon:
1. To find the measure of one interior angle in a regular polygon, first find the sum of interior angles of the required polygon using the formula given below.
Sum of interior angles = (n – 2) ∙ 180°

2. Next, divide the sum of interior angles by the total number of angles the regular polygon has.

To find the measure of each exterior angle in a regular polygon:
The measure of one of the exterior angles of a regular polygon can be found by dividing 360 degrees by the number of angles (n).

$\fn_phv Measure\;of\;one\;exterior\;angle =\frac{360^{\circ}}{n}$

Example

Bees build honeycombs with hexagonal cells. What is the measure of each interior angle of the cell?

Solution

To find the measure of one interior angle in a regular polygon, first find the sum of interior angles of the required polygon using the formula given below.

Sum of interior angles = (n – 2) ∙ 180°

A hexagon has 6 sides, so:
(6 – 2) ∙ 180 = 4 ∙ 180 = 720°

Since this is a regular hexagon, all of the angles are equal, so divide the sum of the interior angles by 6.

$\fn_phv \frac{720^{\circ}}{6}={\color{Red} 120^{\circ}}$

Practice Interior and Exterior Angles of Regular Polygons Word Problems

Practice Problem 1

To manufacture stop signs, the engineers need to determine the measure of each interior angle of the sign. What is the measure of each interior angle if the sign is a regular octagon?

Practice Problem 2

Carl is building a new picnic table for his backyard. The table is in the shape of a regular octagon. What is the measure of each exterior angle of the tabletop?

Practice Problem 3

Marlin is an avid soccer fan and is sewing his own soccer ball. He needs to calculate the angle measure of the shapes on the soccer ball. What is the measure of each interior angle of the regular pentagons?