Interior and Exterior Angles in Polygons - GCSE Maths
Introduction
- The Word Polygon is made up of two words –
- A closed shape made of line segments .
- To make a Polygon, minimum three line segments are required which end up making a Triangle.
- Basic Polygons are Triangle, Square and Rectangle.
- Polygons have vertices, angles and sides.
- An Angle is basically the distance between two rays starting at the same point.
- Polygons have two types of angles, they are – Interior and Exterior angles.
- Polygons are 2-Dimensional shapes and we can use them to make 3-Dimensional objects.
Importance of polygons:
- Polygons play a vital role in understanding geometric concepts like shapes, angles, area and perimeter.
- Polygons are present in our daily life, in buildings, houses, and the design of objects.
- Students learn about angle and length measurements, which are used to solve real-world problems and make maths meaningful.
Types of Polygons
Polygons are classifies into two types –
- Regular Polygons
- Irregular Polygons
Regular Polygons:
- Polygons with equal sides and equal angles.
Irregular Polygons:
- Polygons with unequal sides and unequal angles.
Some Important polygons are as follows-
Interior Angles in Polygons
- The Angles present inside the polygon are known as Interior Angles.
- The polygon with the minimum number of sides is a Triangle and the sum of the interior angles of a Triangle is 180 degree.
- Consider other Polygons divided into triangles –
- We can conclude that every polygon can be divided into triangles, with the number of triangles formed being two fewer than the number of sides of the polygon.
- Since each triangle has interior angles that add up to 180°, the sum of the interior angles of a polygon is given by: –
Where n = Number of sides of the Polygon
Example: The Pentagon has 5 sides so –
- If we want to find the interior angle of a regular polygon, the formula is-
Solved Example:
Solved Example: Problem: Find the missing interior angles in the following Polygon.
The Polygon shown in the diagram is a Hexagon.
In which – 
Solved Example:
Problem: Work out the size of the angle for the following value of n (Number of sides of Regular Polygon).
Solution:
Using formula- 
Put (n = 5)
A Polygon with 5 sides is called a Pentagon.
Divide the sum of angles by number of sides :
Exterior Angles in Polygons
- When we extend any side of a Polygon, then the resulting angle made is called an Exterior Angle.
- When the exterior angles are combined together they form a circle which represents a complete angle of 360 degrees.
- The angles shown below are Exterior angles.
- In the following diagram a Regular Pentagon is shown.
- We know exterior angles summed up together gives us 360 degrees.
- Hence, the relationship between an exterior angle and the sides of the regular polygon, pentagon is-
Solved Example:
Problem: Find the number of sides of the polygon shown in the image given below: 
Step#1: Find the angle of a regular pentagon-
The angle of a regular pentagon will be –
The two angles together make: 
As interior and exterior angles are supplementary –
Solved Example:
Problem: Find out the value of exterior angle x and interior angle y of polygon.
Step#1: Finding the Exterior Angle-
The polygon is an Octagon, and we can find the exterior angle by the formula –
As the Interior and Exterior angles are supplementary, thus-
Solved Example:
Problem: Find the values of the unknown angles.
Step#1: Find the Exterior Angle-
The interior angle of a rectangle is 90°, hence the exterior angle will be-
In triangle A and B, 
In triangle B,
Solved Example:
Problem: Find the value of the unknown angle.
In the diagram, interior angle is 85° and x is unknown.
As they are supplementary-