View source for Angles of Polygons and Regular Tessellations Exploration

{{Exploration}}
{{Time|35}}
{{Objective|Calculate the interior angles of polygons and classify the regular tessellations of the plane.}}
{{clear}}
{{printable|Printable version of this exploration: [[Image:regular-tess-euclid.pdf]]}}
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==Exploration==

===Interior Angles of Polygons===

<ol>

<li> 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?
</li>

<li>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?
</li>
<li>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?
</li>
{| border="1" cellpadding="3"
|-
|[[Image:polygonal-subdivision.svg]]
|align="center"|[[Image:polygonal-subdivision-wrong.svg]]
|-
|align="center"| Cutting polygons into triangles.
|align="center"|A bad way to cut into triangles.
|}


<li>Using the information from question 3 argue that:<br>
<center>The sum of the interior angles of an <math>n</math>-gon  is <math>(n-2)\times 180^\circ</math>
</center>
</li>
<li>Why does the "bad way to cut into triangles" fail to find the sum of the interior angles?
 </li>
</ol>

===Regular Polygons===
A {{define|regular polygon}} is a polygon with all sides the same length and all angles having the same angle measure.

<ol start="6">
<li>  Explain the following formula:
<br>
<center>Each angle of a regular <math>n</math>-gon is <math>\frac{(n-2)180^\circ}{n}</math>.</center>
<br>
Would this formula work for just any <math>n</math>-gon? Why or why not?
</li>
<li> Complete the following table:
{| border="1"
! Number of Sides
|width="5%"|3
|width="5%"|4
|width="5%"|5
|width="5%"|6
|width="5%"|7
|width="5%"|8
|width="5%"|9
|width="5%"|10
|width="5%"|11
|width="5%"|12
|width="5%"|15
|width="5%"|20
|width="5%"|50
|width="5%"|100
|-
! Corner angle = <math>\frac{(n-2)180^\circ}{n}</math>
| 60° || 90° || || || || || || || || || || || ||
|}

<li>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˚?
</li>
<li>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?
</li>
<li>Which regular <math>n</math>-gons are the only ones that can tessellate the plane using just one type of tile?
<li>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°?</li>
</ol>

{{handin}}


[[category:Tessellation Explorations]]