

Interior Angles of a Polygon
Try this
Adjust the polygon below by dragging any orange dot. Click on "make regular" and repeat.
Note the behavior of the interior angles and their sum.
The interior angles of a polygon are those angles at each
vertex that are on the inside of the polygon.
There is one per vertex. So for a polygon with N sides, there are N
vertices and N interior angles.
For a regular polygon,
by definition, all the interior angles are the same. In the figure above, Click on "make regular" then
change the number of sides and resize the polygon by dragging any vertex. Notice that for any given number of sides,
all the interior angles are the same.
For an irregular polygon,
each angle may be different. Click on "make irregular" and observe what happens
when you change the number of sides, and drag a vertex.
Sum of Interior Angles
The interior angles of any polygon always add up to a constant value, which depends only on the number of sides.
For example the interior angles of a
pentagon always add up to 540°
no matter if it regular or irregular,
convex
or
concave,
or what size and shape it is.
The sum of the interior angles of a polygon is given by the formula:
where
n is the number of sides
So for example:
A square 
has 4 sides, 
so interior angles add up to 360° 
A pentagon 
has 5 sides, 
so interior angles add up to 540° 
A hexagon 
has 6 sides, 
so interior angles add up to 720° 
... etc 


In Regular Polygons
For a regular polygon, the total described above is spread evenly among all the interior angles, since they all have the same values. So
for example the interior angles of a pentagon always add up to 540°, so in a regular pentagon (5 sides), each one is one fifth of that, or 108°.
Or, as a formula, each interior angle of a regular polygon is given by:
where
n is the number of sides
Adjacent angles
Two interior angles that share a common side are called "adjacent interior angles" or just "adjacent angles".
While you are here..
... I have a small favor to ask. Over the years we have used advertising to support the site so it can remain free for everyone.
However, advertising revenue is falling and I have always hated the ads. So, would you go to Patreon and become a patron of the site?
When we reach the goal I will remove all advertising from the site.
It only takes a minute and any amount would be greatly appreciated.
Thank you for considering it! – John Page
Become a patron of the site at patreon.com/mathopenref
Other polygon topics
General
Types of polygon
Area of various polygon types
Perimeter of various polygon types
Angles associated with polygons
Named polygons
(C) 2011 Copyright Math Open Reference. All rights reserved

