 #### How do you construct regular polygons?

To construct a regular 3-gon (equilateral triangle), begin with segment AB, and construct two circles AB and BA. The intersection of the two circles at C, will produce an equilateral triangle ABC.

The construct a square, construct a perpendicular bisector of AB and let the point of intersection of the bisector with AB be a new point O, then construct the circle OA. The intersection of the circle with the perpendicular bisector produces points C and D and it should be possible to see that quadrilateral ACBD is square.

The construction of a pentagon is a little more difficult. Begin, as before, bisecting segment AB to locate point O and drawing circle OA, producing points C and D.

Then bisect segment OB to locate point E. By drawing circle ED locate point F on segment AO. The length DF is the required length of the side of an inscribed pentagon. Please note that proof is not provided here, as it requires further proof that cos72o = (√5 – 1)/4, and this would detract too much from polygonal constructions.

The construction of a regular hexagon is relatively simple. We construct a circle OA and length OA is the length of the side of the inscribed hexagon. This is easily demonstrated to be true. As OA is the radius, r, each base length will be r and so we form a series of equilateral triangles. As the centre angle is 60 degrees we have successfully constructed a regular hexagon.

It may be noted that this construction method allows an inscribed equilateral triangle to be drawn, by connecting alternating points.

By combining these constructions with a perpendicular bisector and projecting a ray onto the circle we can double the number of sides.

For example, we can construct a regular octagon from a square. And we can continue this method as often as we wish to produce a 16-gon, 32-gon, etc. Similarly a pentagon allows the construction of the decagon (10-gon), 20-gon, etc. and the hexagon allows us to construct a dodecagon (12-gon), 24-gon, and so on.

The Greeks, naturally, asked the question of other constructions. They knew a useful constructions would be the heptagon (7-gon). In addition they realised that trisecting an angle would be very useful. Instead of doubling sides, this would provide a construction method for tripling the number of sides; for example, an equilateral triangle would give way to a nonagon (9-gon).