Angles of Polygons and Regular Tessellations Exploration

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Objective: There are only three regular tessellations. We will show through some exercises why this is true.

Interior Angles of Polygons

  1. Ho do we know that the sum of the angles in a triangle is 180°? One nice intutitive way to see this is to cut out a triangle. Now tear off the corners and put them together so that their vertices are touching. What do you see?
  2. What is the sum of the angles in a quadrilateral? How do we show that this is true based on the fact that the sum of the angles in a triangle is 180°?
  3. In general, we can divide a polygon up in triangles. (We do have to do this in a clever way.) How many triangles make up a 4-gon? How many triangles make up a 5-gon? How many triangles make up a 6-gon? How many triangles would you say there are in an n-gon based on the pattern described above?
  4. Using the information from question 3 argue that: The sum of the interior angles of an n-gon = (n - 2) 180°
  5. We claimed above that we have to divide up our polygon in a clever way. Why is the following subdivision not appropriate when trying to find the sum of the interior angles?
  6. By definition, all the angles in a regular polygon have the same angle measure. This implies that we get the following formula for the measure of an angle in a regular polygon: The measure of an interior angle in a regular n-gon = (n – 2)180°n Would this formula work for just any n-gon? Why or why not?
  7. Complete the following table:
    n ! Angle = (n – 2) 180° / n
    3 60˚
    4 90˚
    5
    6
    7
    8
    9
    10
    11
    12
    15
    20
    50
    100
  8. If regular polygons are going to fit around a vertex, then their angle measures have to be an exact divisor of 360˚. Which of the angle measures in the table are exact divisors of 360˚?
  9. How do you know that there are no other regular polygons with angles that are exact divisors of 360˚ besides the ones mentioned on the list?
  10. Which regular n-gons are the only ones that can tessellate the plane using just one type of tile?