Number circle for regular polygons and stars

A number circle is a number line wrapped around a circle. Most classrooms have an example on the wall: a clock!

Use the polygon tool (triangle icon, 5th from left) or segment tool (3rd from the left, 2nd on list) to connect the points.

To make a regular n-gon, set the slider to your n (number of equally spaced points on the circle), then connect the points by the "add 1" rule, as in kids' connect the dots coloring books. That is, if you start at 0, connect the points with line segments in this order: 0, 1, 2, 3, 4, 0.

To make a regular n-pointed star polygon, choose another number to add, such as 2. For example, to make a regular 5-pointed star, add 2. Connect the points in this order: 0, 2, 4, 1, 3, 0.

Further details

"Regular" means as symmetric as possible. In this case, it means that all sides are equal and all angles are equal.

For most numbers n, there are many different star polygons possible by following the "add k" rule to connect the points. See if you can find how to predict what you will get:

• For a given n, will every k give an n-pointed star? (Hint: try n=8 and n=9.)
• For a given n, how many different stars can be made? What are the k's that define them?
• For a given n and k, what are the interior and exterior angles at the vertices?
• For which n's will every k give a k-pointed star? (You could think of an n-gon as a star with very blunt points and no crossings.)

Susan Addington, Created with GeoGebra

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