Mechanism for Constructing Regular Polygons

This Demonstration shows a hinged mechanism for constructing regular -gons, . The mechanism is constructed as follows. The fixed hinges are at , , , , , , and , while the hinges at and slide along and respectively, and is the midpoint of . The mechanism contains congruent parallelograms and and two isosceles trapezoids and . This ensures that the sides , , , and have the same length and that angles , , and are equal.

A square is constructed by moving the mechanism to make coincide with . To draw the red perpendicular to at , which is needed in the construction of the pentagon, hexagon, heptagon, and octagon, keep the mechanism rigid and slide it along until a vertical line is at . The nonagon and decagon need the red line to be inclined at 60° and 36°; these angles come from the hexagon and the pentagon.

To construct a heptagon (or 7-gon), keep in place and turn the mechanism until is on the vertical red line. Four sides of the heptagon are determined. Freeze the hinges and rotate the rigid mechanism so that the rod coincides with the previous location of . That determines the remaining three sides. That the pentagon forms one half of the regular heptagon follows from the fact that the polygonal line is symmetric with respect to .