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):
|As with the heptagon, for different orientations, manipulate x- and y-values, as follows:|
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:
- 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
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.