# Angles of Polygons and Regular Tessellations Exploration

From EscherMath

(Redirected from Tessellations: Why There Are Only Three Regular Tessellations)

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

## Exploration

### Interior Angles of Polygons

- 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?
- 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?
- 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 <math>n</math>-gon?
- Using the information from question 3 argue that:

The sum of the interior angles of an <math>n</math>-gon is <math>(n-2)\times 180^\circ</math> - Why does the "bad way to cut into triangles" fail to find the sum of the interior angles?

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

### Regular Polygons

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

- Explain the following formula:

Each angle of a regular <math>n</math>-gon is <math>\frac{(n-2)180^\circ}{n}</math>.

Would this formula work for just any <math>n</math>-gon? Why or why not? - Complete the following table:
Number of Sides 3 4 5 6 7 8 9 10 11 12 15 20 50 100 Corner angle = <math>\frac{(n-2)180^\circ}{n}</math> 60° 90° - 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˚?
- 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?
- Which regular <math>n</math>-gons are the only ones that can tessellate the plane using just one type of tile?
- 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.