Interior Angle Sum Theorem
What is true about the sum of angles inside a polygon (ie interior angles)?
The sum of the measures of the interior angles of a convex polygon with n sides is (n2)180
Examples: Triangle or ( '3gon')
 sum of interior angles: (32) 180 = 180°
 Quadrilateral which has four sides ( ' 4gon')
 sum of interior angles: (42)180 = 360°
 Hexagon which has six sides ( '6gon')
 sum of interior angles: (62)180 = 720°
Video Tutorial
on Interior Angles of a Polygon
Definition of a Regular Polygon: A regular polygon is simply a polygon whose sides all have the same length and whose angles all have the same measure. The most well known example of a regular polygon is the equilateral triangle
Measure of a Single Interior Angle
What about when you just want 1 interior angle?
In order to find the measure of a single interior angle of a regular polygon (a polygon with sides of equal length and angles of equal measure) with n sides, we just divide the sum of the interior angles or (n2) × 180 by the number of sides or n
The FormulaAn interior angle of a regular polygon with n sides is $ \frac{ (n 2) \cdot 180^{\circ} }{n} $
Example:To find the measure of an interior angle of a regular octagon, which has 8 sides, apply the formula above as follows: ((82) × 180) / 8 = 135°
180°
Use Interior Angle Theorem: (5  2) × 180 = 3 × 180 = 540°
Use Interior Angle Theorem: (6  2) × 180 = 4 × 180 = 720°
Finding 1 interior angle of a regular Polygon
Substitute 8 (an octagon has 8 sides) into the formula to find a single interior angle
Substitute 12 (a dodecagon has 12 sides) into the formula to find a single interior angle
Substitute 16 (a hexadecagon has 16 sides) into the formula to find a single interior angle
What is the measure of 1 interior angle of a pentagon?
This question cannot be answered because the shape is not a regular polygon. You can only use the formula to find a single interior angle if the polygon is regular!
Consider, for instance, the irregular pentagon drawn below.
You can tell, just by looking at the picture, that $$ \angle A and \angle B $$ are not congruent.
How about the measure of an exterior angle?
Formula for sum of exterior angles:
The sum of the measures of the exterior angles of a polygon, one at each vertex, is 360°.
Measure of a Single Exterior Angle
Formula to find 1 angle of a regular convex polygon of n sides =
$$ \angle1 + \angle2 + \angle3 = 360° $$
$$ \angle1 + \angle2 + \angle3 + \angle4 = 360° $$
$$ \angle1 + \angle2 + \angle3 + \angle4 + \angle5 = 360° $$
Practice Problems
Substitute 5 (a pentagon has 5sides) into the formula to find a single exterior angle
Substitute 10 (a decagon has 10 sides) into the formula to find a single exterior angle
Substitute 12 (a dodecagon has 12 sides) into the formula to find a single exterior angle
What is the measure of 1 exterior angle of a pentagon?
This question cannot be answered because the shape is not a regular polygon. Although you know that sum of the exterior angles is 360, you can only use formula to find a single exterior angle if the polygon is regular!
Consider, for instance, the pentagon pictured below. Even though we know that all the exterior angles add up to 360 °, we can see, by just looking, that each $$ \angle A and \angle B $$ are not congruent..
Determine Number of Sides from Angles
It's possible to figure out how many sides a polygon has based on how many degrees are in its exterior or interior angles.
Use formula to find a single exterior angle in reverse and solve for 'n'.
Use formula to find a single exterior angle in reverse and solve for 'n'.
Use formula to find a single exterior angle in reverse and solve for 'n'.
If each exterior angle measures 80°, how many sides does this polygon have?
When you use formula to find a single exterior angle to solve for the number of sides , you get a decimal (4.5), which is impossible. Think about it: How could a polygon have 4.5 sides? A quadrilateral has 4 sides. A pentagon has 5 S is nothing in between.

Related:
 Triangles
 Circles
 Quadrilaterals
 images of polygons