Circular Arc Polygon Calculator

Name: Circular Arc Polygon - Geometry Calculator
Author: Jürgen Kummer

Calculations at a circular arc polygon. This lies between regularly round ordered, adjacent circles of the same size, three at least. It is a regular polygon with a circular sector removed from each vertex (for n>3). The side length of the polygon is the diameter of the circles, which is twice the radius.
Enter the number of circles and the circle radius, choose the number of decimal places, then click Calculate.

A circular arc hexagon between six circles in a hexagon.

Formulas:

l = r \cdot (π - \frac{2 π}{n})

h = r \cdot [1 - \cos (\frac{π - \frac{2 π}{n}}{2})]

s = 2 \cdot \sqrt{2 \cdot r \cdot h - h^{2}}

p = n \cdot l

A = n \cdot [\frac{s^{2}}{4 \cdot \tan (\frac{π}{n})} - \frac{r \cdot l}{2} + s \cdot \frac{r - h}{2}]

pi:

π = 3.141592653589793 ...

Radius, lengths, height and perimeter have the same unit (e.g. meter), the area has this unit squared (e.g. square meter).

The first two variants of such a regular circular arc polygon are the circular arc triangle and the circular arc quadrangle. It should be noted that a circular arc triangle between circles of equal size is always regular. This does not apply to shapes with more vertices. Here, the circles can be shifted and still be adjacent. The definition above, that circular arc triangles have circular sectors subtracted from the vertices of the polygon, also does not apply. Instead, circular segments are removed from the sides. The other shapes can also be interpreted as having circular segments removed from their sides. However, in this case, the vertices of the regular polygon are no longer at the centers of the circles, but at the points where the circles touch. This reduces the side length of the polygon.

The symmetry properties of a regular circular arc polygon are the same as those of a regular polygon. It is axially symmetric about a number of axes equal to the number of vertices. With an even number of vertices n, these axes pass through two opposite vertices. With an odd number of vertices, they pass through one vertex and through the midpoint of the opposite side. It is rotationally symmetric about the intersection of all these axes when rotated 360/n degrees and multiples thereof. With an even number of vertices, it is point-symmetric about this intersection point. With an odd number of vertices, it is not point-symmetric.

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