A poly-polygon

(Sorry for the resolution, the file size was too big for my more detailed scan)

The other night I made this shape. As you can see, it is 4 regular polygons inside of each other all sharing the same side length.

I constructed everything with only compass and straightedge, except for the pentagon and hexagon sides coming out of from the base. For the other 2 (or 3) sides of the pentagon/hexagon I copied the angle of the first two lines coming from the base.

Methods used in this construction.

• Copying lengths and angles
• Making an equilateral triangle
• Erecting a perpendicular (for the square, because it has 90 degree interior angles)

To do list

• Explain why pentagons and hexagons have interior angles of 108 and 120 degrees respectively, as well as give proof for a general case for any regular polygon
• Describe how to execute the 3 methods listed above

P.S. Those two purple crayon smudges are driving me crazy.