- Regular polygons.
- Give a definition of a regular polygon.
- How can a regular
*n*-gon be constructed with a ruler and protractor? - What is the measure of an interior angle of a regular
*n*-gon? - Of a central angle?
- Of an exterior angle?

- Angle sums.
- What is the sum of the central angles of a regular polygon?
- Exterior angle sums.
- What is the sum of the exterior angles of a regular polygon?
- Does this result still hold if the polygon is not regular?

- Interior angle sums.
- What is the sum of the interior angles of a regular
*n*-gon? - Does this result still hold if the polygon is not regular but is still convex?
- What if the polygon is not even convex?
- Show how a convex
*n*-gon can be subdivided into*n*-2 triangles. - Can any (not necessarily convex)
*n*-gon be subdivided into*n*-2 triangles?

- What is the sum of the interior angles of a regular

- What might be a good definition for a three-dimensional analogue of a regular polygon?
- 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).- If
*n*=4, what are the coordinates of the other three vertices? - Equilateral triangles.
- If
*n*=3, what are the coordinates of the other two vertices? - Suppose one of these two vertices has coordinates (
*a*,*b*). What is ? Why?

- If
- General regular
*n*-gons.- For general
*n*, what are the coordinates of the other*n*-1 vertices? - Suppose one of these vertices has coordinates (
*a*,*b*). What can you say about powers of*a*+*bi*? Why? - Find all the complex numbers solving ,
- For general
*n*, what is the area of the*n*-gon? - What is the perimeter? Use this to approximate .
- Cut up the polygon into triangles centered at the origin and rearrange them to guess/motivate the formula for the area of a circle.

- For general

- If
- Operations with complex numbers.
- Explain how to add and multiply complex numbers geometrically.
- What is the connection with the angle sum formulas for sine and cosine?
- Give a geometrical interpretation for the square root of -1.
- Find the 3rd roots of
*i*. - Find the 10th roots of
*i*. - How can you find all the
*n*th roots of a general complex number*a*+*bi*?

Wed Nov 4 12:13:22 EST 1998