## Frequently Asked Questions

#### 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 `cos`72^{o} = (√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).