Polygons and Polyhedra

Prepared by:

Joseph Malkevitch
Department of Mathematics and Computer Studies
York College (CUNY)
Jamaica, New York 11451

Email: malkevitch@york.cuny.edu

web page: www.york.cuny.edu/~malk

1. Find the number of degrees for the interior angle of a regular convex polygon with:

a. 3 sides
b. 4 sides
c. 5 sides
d. 6 sides
e. n sides (n at least 3).

2, Can every convex n-gon be subdivided into triangles using the existing vertices for vertices of the triangles? If so, how many triangles are formed?

3. Definition: A diagonal of a convex n-gon is a line which joins two vertices of the n-gon which do not form a side of the polygon.

How many diagonals does a convex 3-gon have? How many diagonals does a convex 4-gon have? How many diagonals does a convex 5-gon have? How many diagonals does a convex n-gon have?

4. Suppose a regular 5-gon has side length 1. Construct a pentagram with the same vertices as the regular 5-gon. What are the the side length and interior angle of the pentagram?

5. Construct as many different ways as you can to subdivide a regular 6-gon into triangles. What seems a reasonable criterion for saying that two such triangulations are the "same" (e.g. isomorphic)? Can you think of another reasonable criterion from the one you chose already? (Hint: You may want to start with a regular 4-gon to think the ideas through?)