Polygons: Formula and Examples
Exterior Angles and Interior Angles
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^{\circ} $
Examples: Triangle or ( '3gon')
 sum of interior angles: $$ (\red 32) 180 = 180^{\circ} $$
 Quadrilateral which has four sides ( ' 4gon')
 sum of interior angles: $$ (\red 42) 180 = 360^{\circ} $$
 Hexagon which has six sides ( '6gon')
 sum of interior angles: $$ (\red 62) 180 = 720^{\circ} $$
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 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{ (\red n 2) \cdot 180^{\circ} }{\red n} $
Example:To find the measure of an interior angle of a regular octagon, which has 8 sides, apply the formula above as follows: $ \frac{(\red82) \cdot 180}{ \red 8} = 135^{\circ} $
180°
Use Interior Angle Theorem:$$ (\red 5 2) \cdot 180^{\circ} = (3) \cdot 180^{\circ}= 540 ^{\circ} $$
Use Interior Angle Theorem: $$ (\red 6 2) \cdot 180^{\circ} = (4) \cdot 180^{\circ}= 720 ^{\circ} $$
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 below.
You can tell, just by looking at the picture, that $$ \angle A and \angle B $$ are not congruent.
The moral of this story While you can use our formula to find the sum of the interior angles of any polygon (regular or not), you can not use this page's formula for a single angle measureexcept when the polygon is regular.
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 sides.

Related:
 Triangles
 Circles
 Quadrilaterals
 images of polygons