Regular N-gon Calculator

Name: Regular N-gon - Geometry Calculator
Author: Jürgen Kummer

Calculations at a regular n-gon or regular polygon.
Enter edge length and number of vertices and choose the number of decimal places. Then click Calculate. The length of a diagonal across a number of edges can also be calculated. Angles are calculated and displayed in degrees, here you can convert angle units.

Regular n-gon shape (up to 100 vertices):

Formulas:

n \in ℕ, n > 2

p = a \cdot n

h = 2 \cdot r_{i} if n is even, else

h = \frac{a}{2 \cdot \tan (\frac{π}{2 \cdot n})}

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

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

r_{i} = \frac{a}{2 \cdot \tan (\frac{π}{n})}

α = 180° - \frac{360°}{n}

d = \frac{n \cdot (n - 3)}{2}

Diagonal across m edges, m∈ℕ, m≤n/2:

d_{m} = a \cdot \frac{\sin (\frac{π \cdot m}{n})}{\sin (\frac{π}{n})}

π = 180° = 3.141592653589793...

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

Another name for the regular polygon is equilateral and equiangular polygon. With an increasing number of corners and sides, it gets closer and closer to the circle and the incircle and circumcircle get closer and closer to each other. With an infinite number of sides, the circle, polygon, incircle and circumcircle are ultimately identical. The length of the sides becomes smaller and smaller as the number increases with the same radius, but with decreasing dimensions. The equilateral triangle is therefore the furthest away from the circle. The most famous is the regular quadrilateral, better known as square.
The regular polygon is point-symmetrical to its center. It is axially symmetric about every perpendicular bisector with an odd number of vertices and with an even number of vertices additionally axially symmetric about every diagonal between two opposite vertices. Therefore, equilateral polygons with n vertices have n axes of symmetry. Rotational symmetry number is identical to the number of corners.
If you remove a similar but smaller polygon from the middle of a regular polygon, you get a polygon ring. The three-dimensional equivalent of regular polygons are regular polyhedra. There are an infinite number of such polygons, but only five polyhedra, the Platonic solids.

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