Part I
Polygons are closed plane figures whose edges are straight lines. Polygons are regular if all of their sides and angles are equal. This means that all the corners, or vertices, of a regular polygon will lie on a circle. Usually the simplest method, then, to construct a regular polygon is to inscribe it in a circle.
Inscribing
an equilateral triangle and a hexagon
Procedure: The radius of a circle can be struck exactly six times around the circle. Connecting the intersections of every other arc yields an equilateral triangle; connecting each successive intersection produces a sixsided figure or hexagon. 
Inscribing
a dodecahedron
Procedure:
Set the compass to the radius of the circle and strike six equidistant
arcs about its perimeter. Connect two neighboring intersections to the
center of the circle. Bisect the resulting angle. Beginning at the intersection
of the bisector and the circle strike six more arcs around the circle.
There will be twelve equidistant intersections on the circle. These will
mark the vertices of a dodecagon.

Inscribing a square
Procedure:
The tilted square, or diamond, was inscribed by connecting the ends
of the horizontal and vertical diameters of a circle. A vertical diameter
can be constructed as the perpendicular bisector of the horizontal diameter
and viceversa. The normal square was inscribed by connecting the diagonal
diameters of the circle. These diameters were constructed by bisecting
the right angles created by the horizontal and vertical diameters.
Inscribing
an octagon
Procedure: Construct horizontal and vertical diameters and then bisect the quadrants of the circle to divide it into eight segments. Connect the endpoints of the four diameters to create an octagon. 
The number of sides of any inscribed polygon may be doubled by further bisecting the segments of the circle. All of polygons above are doublings of the relatively simple constructions of the equilateral triangle and the square. Much more complex are the construction of figures like the pentagon (five sides). This is covered in part II.