Calculations at a regular pentagram or pentacle. A pentagram is constructed from the diagonals of a pentagon. The pentagram is the most simple regular star polygon. The chord slices of a regular pentagram are in the golden ratio φ.
Enter one value and choose the number of decimal places. Then click Calculate.

Formulas:
l = a * φ
b = a / φ
c = b / φ
a = b + c
l = a + b
p = 10 * b
r_{c} = a / 10 * √ 50 + 10 * √5
A = √ 5 * ( 5-2√5 ) * a² / 2

Golden ratio phi:
φ = ( 1 + √5 ) / 2 = 1.618033988749895...

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

The pentagram was already known in Mesopotamia over 5000 years ago. Pythagoras of Samos and the Pythagoreans were particularly interested in it because of the appearance of the golden ratio. Pentagrams were and are used as a symbol for a wide variety of mythological and religious aspects, with two versions to be distinguished: the one with one and the one with two points at the top. Which of course makes no difference mathematically. Both versions were also considered a symbol of protection against evil forces. The national flags of Morocco and Ethiopia contain a pentagram, and they can also be found in some coats of arms, sometimes as a filled star.
The points of the pentagram form a regular pentagon with side length a, inside the pentagram there is a smaller, upside-down pentagon with side length a/φ², with φ as the golden ratio. In this you can of course find another pentagram and so on. The points of the pentagram form isosceles triangles with the side length b and the base length c, whose angle at the apex is 36 degrees and at the base 72 degrees. Accordingly, at the concave corners between two long chord sections b the angle is 108 degrees, as is the convex angle of the inner pentagon and the enclosing outer pentagon.