Polygram Calculator

Name: Polygram - Geometry Calculator
Author: Jürgen Kummer

Calculations at a semi-regular polygram or n-pointed star. This is a regular n-gon with identical isosceles triangles attached to each edge. Please enter at number of points an integer greater as 2. Then, enter the edge length, as well as the base length or one angle. Choose the number of decimal places and click Calculate. Please enter angles in degrees, here you can convert angle units.

Formulas:

b = \sqrt{2 \cdot a^{2} \cdot [1 - \cos (α)]}

α = \arccos (\frac{2 \cdot a^{2} - b^{2}}{2 a^{2}})

β = \frac{360}{n} ° + α

i = \sqrt{\frac{4 \cdot a^{2} - b^{2}}{4}}

l = \sqrt{2 \cdot a^{2} \cdot [1 - \cos (β)]}

p = 2 \cdot n \cdot a

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

pi:

π = 3.141592653589793...

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

A more regular form of the polygram is obtained if equilateral triangles are placed on the sides instead of isosceles triangles. This is achieved by choosing the value of a to be equal to the value of b. This form can be called an equilateral polygram. Only in the case of n=6 is such an equilateral polygram also a completely regular polygram, namely a hexagram.
A polygram with n points has n axes of symmetry. If the number is odd, an axis of symmetry runs from each point through the opposite notch. If the number is even, an axis of symmetry runs from each point through the opposite point and from each notch through the opposite notch. Polygrams with an even number of points are also point-symmetrical to their center. All polygrams are rotationally symmetrical about this center at an angle of 360°/n and multiples thereof. The eight-pointed star chosen as an example above has eight axes of symmetry, four through the points and four through the notches. It is point-symmetric and rotationally symmetric to an angle of 45 degrees and multiples thereof.
The most simple polygrams are threestar and fourstar. A rhombus can be seen as twostar.

Calculation of area and other dimensions: polygram

Last updated on 04/01/2026. Author: Jürgen Kummer

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