The sum of the measures of the interior angles of a convex polygon with n sides is $ (n-2)\cdot180^{\circ} $
| $$ \red 3 $$ sided polygon (triangle) | $$ (\red 3-2) \cdot180 $$ | $$ 180^{\circ} $$ |
| $$ \red 4 $$ sided polygon (quadrilateral) | $$ (\red 4-2) \cdot 180 $$ | $$ 360^{\circ} $$ |
| $$ \red 6 $$ sided polygon (hexagon) | $$ (\red 6-2) \cdot 180 $$ | $$ 720^{\circ} $$ |
What is the total number degrees of all interior angles of a triangle?
180°
You can also use Interior Angle Theorem:$$ (\red 3 -2) \cdot 180^{\circ} = (1) \cdot 180^{\circ}= 180 ^{\circ} $$
What is the total number of degrees of all interior angles of the polygon ?
What is the sum measure of the interior angles of the polygon (a pentagon) ?
What is sum of the measures of the interior angles of the polygon (a hexagon) ?
A regular polygon is simply a polygon whose sides all have the same length and angles all have the same measure. You have probably heard of the equilateral triangle, which are the two most well-known and most frequently studied types of regular polygons.
Regular Hexagon
So, our new formula for finding the measure of an angle in a regular polygon is consistent with the rules for angles of triangles that we have known from past lessons.
Example 2
To find the measure of an interior angle of a regular octagon, which has 8 sides, apply the formula above as follows: $ \text{Using our new formula} \\ \text {any angle}^{\circ} = \frac{ (\red n -2) \cdot 180^{\circ} }{\red n} \\ \frac{(\red8-2) \cdot 180}{ \red 8} = 135^{\circ} $
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 measure--except when the polygon is regular.
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 $$ \angle A \text{ and } and \angle B $$ 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.
If each exterior angle measures 80°, how many sides does this polygon have?
There is no solution to this question.
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.