Angles of Polygons and Regular Tessellations Exploration

Objective: Calculate the interior angles of polygons and classify the regular tessellations of the plane.

Exploration

Interior Angles of Polygons

1. Check that the sum of the angles in a triangle is 180° as follows: Cut out a triangle. Tear off the corners and put them together so that their vertices are touching. What do you see?
2. Draw some quadrilaterals. For each one, show how to cut it into two triangles. Since the angle sum of each triangle is 180°, explain how you know the angle sum of each quadrilateral. What is the angle sum of a quadrilateral?
3. Any polygon can be cut into triangles by connecting its vertices with additional lines. 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 make up an $n$-gon?

Cutting polygons into triangles.	A bad way to cut into triangles.




4. Using the information from question 3 argue that:
   The sum of the interior angles of an $n$-gon is $(n-2)\times 180^{\circ }$
5. Why does the "bad way to cut into triangles" fail to find the sum of the interior angles?

Regular Polygons

A regular polygon is a polygon with all sides the same length and all angles having the same angle measure.

6. Explain the following formula:
   Each angle of a regular $n$-gon is ${\frac {(n-2)180^{\circ }}{n}}$.

   
   Would this formula work for just any $n$-gon? Why or why not?

7. Complete the following table:

   Number of Sides	3	4	5	6	7	8	9	10	11	12	15	20	50	100
   Corner angle = ${\frac {(n-2)180^{\circ }}{n}}$	60°	90°

8. If regular polygons are going to fit around a vertex, then their angle measures have to divide evenly into 360°. Explain. Which of the angle measures in the table divide evenly into 360˚?
9. The table doesn't list every possible number of sides. How do you know that there are no other regular polygons with angles that divide evenly into 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?
11. Three equilateral triangles and two squares can fit together, since 60+60+60+90+90 = 360°. What other combinations of corner angles in the table can be combined to make 360°?

Handin: A sheet with answers to all questions.