Isogonal Polygon Calculator

Name: Isogonal Polygon - Geometry Calculator
Author: Jürgen Kummer

Calculations at an isogonal polygon, or an alternating equiangular polygon. This is a polygon with two alternating edge lengths. Each edge type appears n times, resulting in a total of 2n edges and vertices each. The angles between adjacent edges are uniform and correspond to the angles of a regular polygon. The simplest isogonal polygon is the rectangle, where n=2. With n=3, this is a semiregular -hexagon. The truncated square, an octagon, is an isogonal polygon with n=4.
Enter both edge lengths and number of edge pairs and choose the number of decimal places. Then click Calculate.

Formulas:

p = n \cdot (a + b)

r_{c} = \frac{1}{4} \cdot \sqrt{\frac{{(a + b)}^{2}}{\sin^{2} (\frac{π}{2 n})} + \frac{{(a - b)}^{2}}{\cos^{2} (\frac{π}{2 n})}}

A = \frac{n}{8} \cdot [{(a + b)}^{2} \cdot \cot (\frac{π}{2 n}) - {(a - b)}^{2} \cdot \tan (\frac{π}{2 n})]

pi:

π = 3.141592653589793...

Edge length and perimeter have the same unit (e.g. meter), the area has this unit squared (e.g. square meter).

This formula for the area of the isogonal polygon has not been formally proven. It was derived by treating the polygon not as a surface, but as a sequence of edges. Each edge has a known length and is rotated by a constant exterior angle of π/n relative to the preceding one. The coordinates of all vertices can be calculated from these edges. Subsequently, Gauss's polygon area formula (also known as the shoelace formula) is applied to determine the area directly from the vertex coordinates. Due to the polygon's regular structure, the resulting sums simplify into geometric series that can be evaluated in closed form. After some trigonometric manipulations, the compact area formula shown above is obtained. It was subsequently verified through independent calculations using Gauss's polygon area formula for various values of a, b, and n.
The formula for the circumcircle radius can be derived directly by decomposing the polygon into two superimposed regular 2n-gons or from the explicit vertex representation. This derivation is simpler than that of the area formula.
At the time of writing, these formulas are not confirmed in accessible literature.

↑ up