
How to construct an arbitrary regular Nsided polygon, inscribed in a given circle?


As stated in the constructions page, there are many regular polygons that cannot be constructed by the rulerandcompass rule. One example is the heptagon. The given X and Ycoordinates of the angular points of the heptagon are used to draw it as accurately as needed. This is a special case.
In the general case, the X and Ycoordinates are derived from the cosine and sine of the angle made by the angular point relative to the (positive) horizontal direction. If r denotes the radius of the circle, and N the total number of angular points, then the coordinates of the ith angular point (i = 0, 1, ..., N1) are:
X = r * cos (i/N*360°)
Y = r * sin (i/N*360°)
With a scientific calculator (for instance the Windows calculator in scientific mode), these values can simply be obtained.
As an example, here are the coordinates of a regular 13sided polygon, inscribed in a circle with radius = 1.0000, and pointing to the right (calculator in the "DEG"mode): 
i  i/N*360°  X  Y 
 0  0.0000°  1.0000  0.0000 
 1  27.6923°  0.8855  0.4647 
 2 
55.3846°  0.5681 
0.8230 
 3 
83.0769°  0.1205 
0.9927 
 4 
110.7692°  –0.3546 
0.9350 
 5 
138.4615°  –0.7458 
0.6631 
 6 
166.1538°  0.9709 
0.2393 
 7 
193.8462°  –0.9709 
–0.2393 
 8 
221.5385°  0.7458 
–0.6631 
 9 
249.2308°  –0.3546 
–0.9350 
 10 
276.9230°  0.1205 
–0.9927 
 11 
304.6154°  0.5681 
–0.8230 
 12 
332.3077°  0.8855 
–0.4647 

As with the heptagon, for different orientations, manipulate x and yvalues, as follows:
 for the polygon pointing to the left, exchange + and – of all values
 for the polygon pointing up, exchange x and yvalues
 for the polygon pointing down, do both, exchange + and –, and exchange x and yvalues
If an arbitrary orientation is needed, where the starting angular point makes an angle say α with the horizontal direction, this angle must be added to the angles given above. In this case, the coordinates of the ith angular point become:
X = r * cos (i/N*360° + α)
Y = r * sin (i/N*360° + α)
The given coordinate values assume a circumscribed circle with radius 1.0000. For polygons of different sizes, multiply all x and yvalues with the desired radius of the circumscribed circle, according to the given formulas for X and Y above.


