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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) ?

Answer The sum of the measures of the interior angles of a convex polygon with n sides is (n-2)180

    Examples:
  • Triangle or ( '3-gon' )
  • Quadrilateral which has four sides ( ' 4-gon')
    • sum of interior angles: (4-2)180 = 360°
  • Hexagon which has six sides ( '6-gon')
    • sum of interior angles: (6-2)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.

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


What is the total number of degrees of all interior angles of the polygon on the left?
Answer


What is the sum measure of the interior angles of the polygon (a pentagon) on the left?
Answer


hexagon What is sum of the measures of the interior angles of the polygon (a hexagon) on the left?
Answer
 
Finding 1 interior angle of a regular Polygon
What is the measure of 1 interior angle of a regular octagon?
Answer


Calculate the measure of 1 interior angle of a regular dodecagon (12 sided polygon)?
Answer


Calculate the measure of 1 interior angle of a regular hexadecagon (16 sided polygon)?
Answer


challenge problem Challenge Problem)



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 angleA and angleB are not congruent.
pentagon irregular

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 = a


Exterior Angles of Triangle


1+2 +3 =360°



Exterior Angles of Polygon


1+2 +3+ 4 =360°


Exterior Angles of Pentagon


41+32 +23+ 1 4 +5 =360°
Practice Problems

Calculate the measure of 1 exterior angle of a regular pentagon?
Answer
What is the measure of 1 exterior angle of a regular decagon (10 sided polygon)?
Answer
What is the measure of 1 exterior angle of a regular dodecagon (12 sided polygon)?
Answer
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 angleA and angleB 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?
Answer
problem 2) If each exterior angle measures 20°, how many sides does this polygon have?
Answer
problem 3) If each exterior angle measures 15°, how many sides does this polygon have?
Answer
Challenge Problem) If each exterior angle measures 80°, how many sides does this polygon have?
Answer