Angles of Polygons and Regular Tessellations 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 n-gon?
- Using the information from question 3 argue that:
The sum of the interior angles of an n-gon is
- 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.|
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 n-gon is .
Would this formula work for just any n-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 = 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 n-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°?