Star Shape Calculator

What Is a Star Polygon?

A star polygon (star shape) is a non-convex, self-intersecting, equilateral, and equiangular polygon. Common examples include the pentagram (5-pointed star), hexagram (Star of David), heptagram, and octagram. Each star is identified by a Schläfli symbol $n/m$ where $n$ is the number of points and $m$ is the starriness.

Explore related geometry tools: Regular Polygon Calculator and Pentagon Calculator.

Pentagram Formulas

For a pentagram with point distance $a$ and golden ratio $\varphi = \frac{1+\sqrt{5}}{2}$:

$$l = a\varphi, \quad b = \frac{a}{\varphi}, \quad c = \frac{b}{\varphi}, \quad P = 10b$$

$$A = \frac{\sqrt{5(5-2\sqrt{5})}}{2} \cdot a^2$$

Hexagram Formulas

For a hexagram: $b = \frac{a}{\sqrt{3}}$, $c = b$, $l = 3b$, $P = 12b$, and $A = 3b^2\sqrt{3}$.