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