# 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**

- 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?
- 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°?
- 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?
- Using the information from question 3 argue that: The sum of the interior angles of an n-gon = (n - 2) 180°
- 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?
- 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?
- 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 - 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˚?
- 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?
- Which regular n-gons are the only ones that can tessellate the plane using just one type of tile?