Approximate Construction of Heptagon and Nonagon

Avni Pllana

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A simple approximate construction of heptagon is shown in Fig.1.

Fig.1

First we construct points C = 1/2 and D = 1/5 on the x-axis of the unit circle, as shown in Fig.1. Then we construct point E = 1/2( C + D ). The line BE intersects the circle at point F. Now we have

angle AOF = 51.4199 .

The angle subtended by a side of a regular heptagon is 360 / 7 = 51.4286 . So angle AOF is about 0.0087 off, which can be considered as a good approximation.

 

Another more accurate construction of heptagon is shown in Fig.2.

Fig.2

The arc with center at B and radius OA = 1 intersects the unit circle at point C. On the line BC we construct point D such that CD = 1/3CB. The arc with center at E = -1/5 and radius EA intersects the y-axis of the unit circle at point F. The line FD intersects the circle at point G. Now we have

angle AOG = 51.4289 ,

which can be considered as a very good approximation.

 

An approximate construction of nonagon is shown in Fig.3.

Fig.3

The arc with center at A and radius OA = 1 intersects the unit circle at point B. We construct point C such that C = 1/4 +1/5. The arc with center at B and radius BC intersects the circle at point D. Now we have

angle AOD = 40.0059 .

The angle subtended by a side of a regular nonagon is 360 / 9 = 40 , so angle AOD can be considered as a good approximation.

 

Another more accurate construction of nonagon is shown in Fig.4.

Fig.4

The arc with center at A and radius OA = 1 intersects the unit circle at point C. On the line AC we construct point D such that CD = 1/3CA. The arc with center at O and radius OA = 1/2 intersects the bisector of the second quadrant at point E. On the negative y-axis we construct point B = -a12, where a12 is the side of the regular 12-gon (dodecagon). The line BE intersects the x-axis at point F. The line FD intersects the circle at point G. Now we have

angle AOG = 40.0002 ,

which can be considered as a very good approximation.

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