A guide to the construction of regular polygons 3-10 (2024)(exact constructions)

The construction of regular polygons has been entertaining humans for quite a long time now and after thousands of years most of us haven't got very far with getting a clear understanding about the structure of regular polygons never mind the irregular ones.

Here is my take on the construction of  regular polygons 3-10 , on 23 Feb 2024:

3-4 have been known now for quite a long time (constructible with compass and ruler).

A line , two points and circles and you have the defining lines/points of a triangle and a square.

This also defines the square root of 2 and 1/2 square root of 3.

 

Construction of a triangle and a square, the lines of the square root 2 and 1/2 square root 3 are also shown

5 or the pentagon gets a lot of literature as polygons go. The golden ratio gets a lot of literature too.

We can find the golden ratio in the pentagon diagonals, that is a way that we can construct a pentagon with compass and ruler.

Let's look first at the golden ratio/ square root 5.

If the sides of the square have a unit of one, the line in red has a value of square root 5.

If the sides of the square have a unit of one the red line has the value of the golden ratio also known as phi.

If the base of the pentagon is one, the line from the bottom side of the pentagon to the top is phi or the golden ratio.

6, the hexagon and first of what I call the doubles (4n+2)

I call the doubles to polygons that share parallel lines .

All the lines of a hexagon are defined by a triangle.

Extended lines of a triangle

A hexagon constructed with extended lines/circles of a triangle.

7, or heptagon

A heptagon can not be constructed by compass and ruler (the first polygon to not do so, it didn't take long).

A heptagon can however be constructed by compass and marked ruler (a neusis construction) . 

Square root 2 and 1  , and you get the heptagon

8, 4n , octagon

The octagon in red, the square in green the square root 2 in blue, that's the 4n's.

One of the easiest way to build an octagon  is to start with a square and find the centre point. Draw circles  from the corners of the square to the centre of the square and there is the octagon(compass and ruler).

9, nonagon (the nonagon can not be constructed by compass and ruler)

The nonagon (6n+3),in the similar way as the heptagon can be constructed by compass and marked ruler (neusis construction).In the case of the nonagon we start with the hexagon as a base and concentrate on the middle line of value 2 in the centre (considering that the base of the hexagon is one).

Another line of 2 will meet the middle line of the hexagon from the top of the nonagon. This line runs down to the side of the nonagon/hexagon. 

10, decagon

The decagon is a double (4n+2), and the lines of a pentagon will define the decagon.