

Polygons: Formula and ExamplesExterior Angles and Interior AnglesThis Page: Definition of Regular Polygon  Exterior Angle of Polygon Formula  Interior Angle of Polygon Formula
Answer
The sum of the measures of the interior angles of a convex polygon with n sides is (n2)180
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
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 (n2) × 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:
What is the measure of 1 interior angle of a regular octagon?
Substitute 8 (an octagon has 8 sides) into the formula to find a single interior angle Calculate the measure of 1 interior angle of a regular dodecagon (12 sided polygon)?
Substitute 12 (a dodecagon has 12 sides) into the formula to find a single interior angle Calculate the measure of 1 interior angle of a regular hexadecagon (16 sided polygon)?
Substitute 16 (a hexadecagon has 16 sides) into the formula to find a single interior angle Challenge Problem) What is the measure of 1 interior angle of a pentagon ?
Exterior Angles of a PolygonFormula for sum of exterior angles:The sum of the measures of the exterior angles of a polygon, one at each vertex, is 360°. 1+2 +3 =360° 1+2 +3+ 4 =360° 1+2 +3+ 4 +5 =360°
Calculate the measure of 1 exterior angle of a regular pentagon?
Substitute 5 (a pentagon has 5sides) into the formula to find a single exterior angle
What is the measure of 1 exterior angle of a regular decagon (10 sided polygon)?
Substitute 10 (a decagon has 10 sides) into the formula to find a single exterior angle
What is the measure of 1 exterior angle of a regular dodecagon (12 sided polygon)?
Substitute 12 (a dodecagon has 12 sides) into the formula to find a single exterior angle
Challenge Problem) What is the measure of 1 exterior angle of a pentagon?
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?
problem 2) If each exterior angle measures 20°, how many sides does this polygon have?
problem 3) If each exterior angle measures 15°, how many sides does this polygon have?
Challenge Problem) 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 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. 