Free Star Shape Calculator

Star Polygon Calculator: A Online Geometry Tool for Star Dimensions

The star shape calculator (also referred to as a pentagram calculator or hexagram calculator) is a free online utility dedicated to the geometry of regular star polygons. Whereas standard polygon calculators handle convex shapes, this tool focuses on self‑intersecting stars—specifically pentagrams, hexagrams, heptagrams, and octagrams. Enter any single known value (side length, perimeter, or area) and the calculator instantly derives all corresponding measurements.

Fundamentals of Star Polygons

A star polygon is defined as a non‑convex, self‑intersecting shape that is equilateral and equiangular. Every regular star polygon is identified by the Schläfli symbol $\{n/m\}$:

• $n$ = number of vertices (and sides) of the underlying regular polygon.
• $m$ = starriness, or the number of times you must travel around the center to return to the starting point.

This notation neatly classifies all possible star polygons that can be constructed from an $n$-gon. Because star polygons are self‑intersecting, their area calculation differs from that of convex polygons; the calculator uses exact formulas for each shape.

Pentagram: The Five‑Pointed Star

The pentagram is the smallest regular star polygon, derived from a pentagon. Its line segments are in a golden ratio relationship, making it a favorite subject in both art and geometry.

Key Variables

In the star calculator, the following measurements are used:

• $a$ – distance between two adjacent tips.
• $b$ – length of each ray from a tip to the first intersection (the “point” of the star).
• $c$ – side of the inner convex pentagon.
• $l$ – full width along one side of the star (tip‑to‑tip across that side).

Golden Ratio Relations

For a pentagram, the segments follow the golden ratio $\phi \approx 1.618$:

$$b = \frac{a}{\phi}, \quad c = \frac{b}{\phi} = \frac{a}{\phi^{2}}, \quad l = \frac{c}{\phi} = \frac{a}{\phi^{3}}$$

Alternatively, in terms of the ray length $b$:

$$a = b \times \phi, \quad c = b \times \phi^{2}, \quad l = b \times \phi^{3}$$

Perimeter and Area

The total length of the outer boundary (the sum of all ten ray segments) is

$$P = 10 \times b$$

The area of a pentagram is a more involved expression; the calculator uses the formula

$$A = \frac{5}{2} \cdot b^{2} \cdot \frac{\sin(2\pi/5)}{\sin(\pi/5)}$$

which simplifies to a closed‑form value involving $\phi$.

Hexagram: The Six‑Pointed Star

Built from a regular hexagon, the hexagram consists of two interlocking equilateral triangles. This symmetry introduces the square root of three into many geometric relationships. For example, the distance between opposite tips and the ray length are linked by a factor of $\sqrt{3}$. The tool can compute the area directly, using a component formula such as

$$A = 3\sqrt{3} \cdot b^{2}$$

(where $b$ is the ray length) for certain decompositions.

Heptagram: The Seven‑Pointed Star

A regular heptagon gives rise to two distinct star polygons: the $\{7/2\}$ and $\{7/3\}$ heptagrams. The $\{7/2\}$ form has a wider appearance, while the $\{7/3\}$ one is more slender. Both can be analyzed by dividing the star into a central convex heptagon and seven surrounding isosceles triangles. The calculator supports both Schläfli symbols, allowing you to input one known dimension for each type.

Octagram: The Eight‑Pointed Star

The octagram originates from a regular octagon and also offers two variants: $\{8/2\}$ and $\{8/3\}$. Angles of $45^{\circ}$ appear frequently, making $\sqrt{2}$ a common factor in the formulas. Its area can be obtained by combining a central octagon with eight flanking triangles.

Fascinating Facts: Star Polygons in the World

Star‑shaped polygons appear on many national flags. The United States flag contains 50 pentagrams; Israel and Burundi use hexagrams; Jordan features a heptagram; and Azerbaijan proudly displays an octagram. The pentagram also appears in religious and occult symbolism, though its geometric appeal lies in its golden ratio proportions.

A classic puzzle asks: “How many triangles are hidden inside a pentagram?” The answer is 35 when counting all overlapping triangles formed by the star’s intricate pattern.

How to Use the Star Dimensions Calculator

1. Pick the desired star type from the menu (pentagram, hexagram, heptagram, or octagram).
2. Enter one known measurement: any of the linear parameters ($a$, $b$, $c$, $l$), the perimeter, or the area.
3. The tool instantly fills in all other fields.
   • For heptagrams and octagrams, beginning with a linear dimension is recommended because the area formulas involve the cotangent function, which prevents straightforward inversion from area alone.

All calculations are based on exact geometric formulas, ensuring both accuracy and consistency across parameters.