Regular Polygon

How to construct an arbitrary regular N-sided polygon, inscribed in a given circle?

As stated in the constructions page, there are many regular polygons that cannot be constructed by the ruler-and-compass rule. One example is the heptagon. The given X- and Y-coordinates 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 Y-coordinates 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 i-th angular point (i = 0, 1, ..., N-1) 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 13-sided polygon, inscribed in a circle with radius = 1.0000, and pointing to the right (calculator in the "DEG"-mode):
ii/N*360°XY
00.0000°1.00000.0000
127.6923°0.88550.4647
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 y-values, as follows:
  • for the polygon pointing to the left, exchange + and of all values
  • for the polygon pointing up, exchange x- and y-values
  • for the polygon pointing down, do both, exchange + and , and exchange x- and y-values
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 i-th 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 y-values with the desired radius of the circumscribed circle, according to the given formulas for X and Y above.

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Copyright © 2017, Zef Damen, The Netherlands