Round-Edged Polygon Calculator

Name: Round-Edged Polygon - Geometry Calculator
Author: Jürgen Kummer

Calculations at a round-edged polygon. This is a regular n-gon with matching equal circular segments attached to each edge.
Enter the number of edges of the polygon, respectively the number of circular segments of the round-edged polygon and one value of arc length, circle radius, segment height or edge length. Choose the number of decimal places, then click Calculate.

Formulas:

Polygon angle

α = π - \frac{2 π}{n}

l = r \cdot α

h = r \cdot [1 - \cos (\frac{α}{2})]

h = \frac{a}{2} \cdot \tan (\frac{α}{4})

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

p = n \cdot l

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

pi:

π = 3.141592653589793 ...

Lengths, radius, height and perimeter have the same unit (e.g. meter), the area has this unit squared (e.g. square meter). The circle radius is the radius of the circle of the segments.

The round-edged polygon creates the impression of a highly simplified flower. With this shape, the number of vertices determines the angle of the circular segment and, consequently, the shape of the petals. The more petals there are, the flatter they become. This method allows such a shape to be fully described using just two parameters. However, if the petals, or circular segments, are to have arbitrary heights, at least one additional parameter is required to decouple the circular segment from the polygon's vertex angle. The calculation becomes even more complex if the petals are based on a more intricate shape, such as an ellipse, rather than a circle.

Such a round-edged polygon is non-convex, all its vertices are concave. It has no straight edges. In contrast, a convex shape is created by rounding the vertices of a polygon. If this is done uniformly, the result is a rounded regular polygon, a shape with no vertices but with straight edges.

This round-edged polygon shares the same symmetry properties as a regular polygon. Axial symmetry passing through each original edge and the opposite vertex or edge, point symmetry if there is an even number of vertices, and rotational symmetry involving rotations of 360°/n and multiples thereof.

↑ up