Regular Polygon Ring Calculator

Name: Regular Polygon Ring - Geometry Calculator
Author: Jürgen Kummer

Calculations at a regular polygon ring, a polygon with a hole. This is a regular n-gon, of which a smaller, similar n-gon was removed from its center.
Enter the edge lengths and the number of vertices of one n-gon and choose the number of decimal places. Then click Calculate.

Formulas:

n \in ℕ, n > 2

c = \frac{a - b}{2 \cdot \sin (\frac{π}{n})}

d = \frac{a - b}{2 \cdot \tan (\frac{π}{n})}

p = (a + b) \cdot n

A = \frac{n \cdot (a^{2} - b^{2})}{4 \cdot \tan (\frac{π}{n})}

pi:

π = 3.141592653589793...

Edge lengths, thicknesses and perimeter have the same unit (e.g. meter), the area has this unit squared (e.g. square meter). Perimeter refers to inner and outer boundary lines.

The polygon ring is axially symmetric, and the number of its axes of symmetry is equal to the number of vertices of one of the two underlying polygons. With an even number of vertices, each axis of symmetry passes through an outer vertex and the opposite outer vertex, as well as the corresponding inner vertices. With an odd number of vertices, the axis of symmetry runs from a pair of vertices through the center of the opposite side. Polygon rings with an even number of vertices are point-symmetric about their center. Each polygon ring is rotationally symmetric about its center when rotated by 360 degrees divided by the number of outer (or inner) vertices.
The surface area of the polygon ring is, of course, the surface area of the larger polygon minus that of the smaller, removed polygon. The perimeter is defined here as the sum of the perimeters of these two shapes. If perimeter refers only to the outer boundary, then this is the perimeter of the outer polygon, with the formula a * n.

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