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.
What about when you just want 1 interior angle?
Measure of a Single Interior Angle
Answer:
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 (n-2) × 180 by the number of sides or n
The Formula
An 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:
( (8-2) × 180) /8 = 135°
What is the total number degrees of all interior angles of a triangle?
Answer
180°
What is the total number of degrees of all interior angles of the polygon on the left?
Answer
360° since the polygon on the left is really just two triangles and each triangle has 180°.
What is the sum measure of the interior angles of the polygon (a pentagon) on the left?
What is the measure of 1 interior angle of a pentagon ?
Answer
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 A and B are not congruent.
How about the measure of an exterior angle?
Exterior Angles of a Polygon
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 =
1+2 +3 =360°
1+2 +3+ 4 =360°
1+2 +3+ 4 +5 =360°
Practice Problems
Calculate the measure of 1 exterior angle of a regular pentagon?
Challenge Problem) What is the measure of 1 exterior angle of a pentagon?
Answer
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 A and 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.
problem 1) If each exterior angle measures 10°, how many sides does this polygon have?
Challenge Problem) If each exterior angle measures 80°, how many sides does this polygon have?
Answer
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 abou it: How could a polygon
have 4.5 sides? A quadrilateral has 4 sides. A pentagon has 5 sides...there is nothing in between.
Challenge Problem)
1+
