Regular Polygon

Each regular polygon has a center. This center can be found by constructing the perpendicular bisectors of any two sides of the regular polygon that are not opposite each other. The point of intersection of the perpendicular bisectors is the center of the regular polygon. For polygons with an even number of sides, the center can be found by connecting any two sets of antipodal (opposite) points.

The central angle of a regular polygon is the angle between two rays that go from the center of the regular polygon and pass through two adjacent vertices of the polygon. The measure of the central angle of regular polygons is or where n is the number of vertices of the polygon.

A circle can be drawn around every regular polygon that intercepts all the vertices of the polygon and none of the sides. This is the circumcircle of the regular polygon. The center of the regular polygon is also the circumcenter of the regular polygon.

To construct the circumcircle about a regular polygon, place the point of the compass at the center of the polygon, and the stylus on a vertex, then draw the circle.

A circle can be drawn inside every regular polygon that intercepts each of the sides of the polygon exactly once. This is the incircle of the regular polygon. The center of the regular polygon is also the incenter of the regular polygon.

To construct the incircle of a regular polygon, construct the midpoint of any of the sides. Then place the point of the compass at the center of the polygon, and the stylus on the midpoint, then draw the circle.