Explorations with Polygons

1. Regular polygons.
   1. Give a definition of a regular polygon.
   2. How can a regular n-gon be constructed with a ruler and protractor?
   3. What is the measure of an interior angle of a regular n-gon?
   4. Of a central angle?
   5. Of an exterior angle?
2. Angle sums.
   1. What is the sum of the central angles of a regular polygon?
   2. Exterior angle sums.
      1. What is the sum of the exterior angles of a regular polygon?
      2. Does this result still hold if the polygon is not regular?
   3. Interior angle sums.
      1. What is the sum of the interior angles of a regular n-gon?
      2. Does this result still hold if the polygon is not regular but is still convex?
      3. What if the polygon is not even convex?
      4. Show how a convex n-gon can be subdivided into n-2 triangles.
      5. Can any (not necessarily convex) n-gon be subdivided into n-2 triangles?
3. What might be a good definition for a three-dimensional analogue of a regular polygon?
4. Inscribe a regular n-polygon in a circle of radius one centered at the origin, with one vertex of the polygon at the point (1,0).
   1. If n=4, what are the coordinates of the other three vertices?
   2. Equilateral triangles.
      1. If n=3, what are the coordinates of the other two vertices?
      2. Suppose one of these two vertices has coordinates (a,b). What is ? Why?
   3. General regular n-gons.
      1. For general n, what are the coordinates of the other n-1 vertices?
      2. Suppose one of these vertices has coordinates (a,b). What can you say about powers of a+bi? Why?
      3. Find all the complex numbers solving ,
      4. For general n, what is the area of the n-gon?
      5. What is the perimeter? Use this to approximate .
      6. Cut up the polygon into triangles centered at the origin and rearrange them to guess/motivate the formula for the area of a circle.
5. Operations with complex numbers.
   1. Explain how to add and multiply complex numbers geometrically.
   2. What is the connection with the angle sum formulas for sine and cosine?
   3. Give a geometrical interpretation for the square root of -1.
   4. Find the 3rd roots of i.
   5. Find the 10th roots of i.
   6. How can you find all the nth roots of a general complex number a+bi?