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5.1 – Angle Sums of a Polygon

Key Terms

  • Concave – Having one or more indentations.
    • In math, a polygon is concave if a line can be drawn that contains a side of the polygon and also contains points inside the polygon.
  • Convex – Having no indents.  No lines can be drawn between vertices that are inside of the polygon.
  • Decagon – A polygon with 10 sides.
  • Diagonal – A line segment that connects two nonconsecutive vertices of a polygon.
  • Exterior Angles – Angles that are on the outside of a polygon and form linear pairs with interior angles of the polygon.
  • polygon – A closed plane figure.
  • Regular Polygon – A convex polygon in which all sides and angles are congruent.



  • The word polygon comes for the Greek prefix poly-,meaning many, and the Greek root gonia, which means angle.
    • So, a polygon is a shape with many angles!
  • A vertex of a polygon is the point where two of its sides meet.
    • The plural of vertex is vertices
  • Triangles are 3 sided polygons




  • Polygons
    • Two-dimensional (plane)
    • Closed (no openings in sides)
    • All straight sides (segments)
    • At least three sides
    • At least three angles
    • At least three vertices
  • Name polygons in clockwise and counter-clockwise order.

5.1 - NamingPolygons


  • Concave and Convex Polygons
    • Click the “play” button to watch the video, then go to Pg 2 in Apex Study 5.1 to try sorting the rest of the polygons.
  • Concave polygons have dents, and you can draw a line through them when you trace a side.

5.1 - Concave

  • Regular Polygons
    1. Convex
    2. Equilateral (same side lengths, congruent side lengths)
    3. Equiangular (same interior angle measures and same exterior angle measures)

5.1 - Regular Polygons



  • Click the “play” button to watch the video, then go to Pg 4 in Apex Study 5.1 to try match the names of common polygons to their side lengths.


  • Diagonals
    • The word diagonal comes from the Greek prefixdia-, meaning “across,” and the Greek root gonia, which means “angle.”
    • A diagonal goes across a polygon from angle to angle
    • Formula for Finding the Number of a Polygon with “n” Sides: n(n − 3) / 2


  • If you draw all of the diagonals from ONE vertex, you can divide the polygon into triangles

5.1 - DiagonalsOneVertex

  • Each triangle’s interior angles add up to 180°.
  • How many triangles do you see in each polygon above?
    • To find the SUM of the Interior Angles of a Polygon, multiple the number of triangles by 180°, or use this formula:
    • (n-2) * 180 = Sum of Interior Angles of a Polygon
      • Why n-2?  Well, each n is a side.  So, using the polygons above, notice the first one has 4 sides.  And, 4 – 2 = 2.  Do you see 2 triangles?  2 * 180 = 360.
      • In the second polygon, you count 5 sides.  5 – 2 = 3.  Do you see 3 triangles?  3 * 180 = 540.
      • In the third polygon, you count 8 sides.  8 – 2 = 6.  Do you see 6 triangles?  6 * 180 = 1080.

5.1 - Angle Sums Triangles



  • Sum of Interior Angle Formula (learned with Diagonals, above)



  • This table will come in handy as a quick reference guide



  • Regular Polygons have CONGRUENT Interior Angles
    • To find the measure of ONE interior angle of a regular polygon, find the SUM of the Interior Angles, then divide by how many angles (vertices) there are!
    • Ex. A Heptagon has 7 sides and the sum of interior angles is 900 (see chart above).
      • Since it has 7 sides, it also has 7 angles.
      • 900 divided by 7 = 128.57°.
      • So, each angle of a regular heptagon is 128.57 degrees!


  • Play the game on Apex Study 5.1, Pg 15 to get some practice!


  • Exterior Angles of a Polygon
    • Add up to 360°
    • Are on the OUTSIDE of the polygon
    • Form LINEAR PAIRS with Interior Angles of a polygon
    • Are supplementary (180°) with each interior angle of the polygon
    • There are 2 exterior angles for every vertex of a polygon
    • They are NOT vertical from interior angles
    • They are NEVER obtuse!

5.1 - Angle Sums Exterior

5.1 - Angle Sums Exterior



  • Formula for Finding Exterior Angles of a Polygon (written 2 ways, because multiplication and division are inverses)
    • 360 / n = Exterior Angle Measure
    • Exterior Angle Measure * n = 360
    • Click “play” to see an example of how to find the Sum of Interior Angles of a polygon if you only know the measure of ONE exterior angle.
  • Try the practice problems in Apex 5.1 on Pg 25.


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