Twinkle, twinkle, little star... our star shape calculator will calculate all of the important elements of a star polygon in the blink of an eye! Here you will learn everything about star-shaped polygons — well, everything apart from occultism: we deal with geometry, not magic!
Keep reading to find out what star-shaped polygons are, how to build them, what a pentagram really is, which other star shapes exist, and how to use our calculator. We even added some fun facts and a little challenge along the way. To the stars! 🌟
Star shapes: pointy geometry
Star polygons are the shiniest shapes in geometry. By definition, a regular star polygon is:
- Non-convex — you can draw a line between two points inside the shape that passes outside its perimeter.
- Self-intersecting — you can build every star shape by starting from a regular polygon and prolonging its sides until the extensions cross each other.
- Equilateral — all the outer sides have the same length (latus is Latin for "side").
- Equiangular — all the angles lying in the same "region" are equal.
The last two attributes automatically qualify star polygons as regular polygons. You can of course draw irregular stars, but those would be impossible to describe with a single formula — and that is exactly what this calculator is here for!
The Schläfli symbol
We can uniquely identify every star polygon using the Schläfli symbol, written as a pair of numbers {n/m}, where:
- n is the number of corners (points) of the star; and
- m is its starriness (density) — how many distinct boundaries you can identify around the center.
To build a star polygon, place n points evenly on a circle and connect every m-th point with a straight line until you return to the start.
What is the pentagram shape?
The five-pointed star — the pentagram — has the Schläfli symbol {5/2}. It is so important that it deserves its own section. You obtain it by taking five points on a circle and joining every second one, tracing all five lines in a single continuous stroke.
Inner radius: r = R × cos(mπ/n) / cos((m−1)π/n)
Perimeter: P = 2n × edge Area: A = n × R × r × sin(π/n)
For the pentagram {5/2}, the point angle is 36° and the inner pentagon has a vertex angle of 108°. The famous golden ratio, φ ≈ 1.618, appears everywhere in a pentagram!
More than pentagram: different star shapes
The pentagram is only the beginning. By changing n and m you obtain a whole family of stars:
- {5/2} — pentagram (five-pointed star).
- {6/2} — hexagram, the Star of David (a compound of two triangles).
- {7/2} and {7/3} — the two heptagrams (seven-pointed stars).
- {8/3} — octagram (eight-pointed star).
- {n/m} — in general, as long as 2 ≤ m < n/2.
The larger the density m, the sharper and pointier the star becomes; the smaller it is, the closer the shape gets to a plain convex polygon.
Some facts about star shapes!
- 💡 A star polygon {n/m} and {n/(n−m)} describe the same shape, so we only need m < n/2.
- 💡 How many triangles are in a pentagram? The classic answer is 35 — count the small ones, the medium ones, and the large ones!
- 💡 A pentagram divides its own sides in the golden ratio, again and again.
- 💡 When m does not divide evenly into n (their greatest common divisor is 1), the star can be drawn in one continuous line without lifting your pen.
How to use our star shape calculator
- Pick your unit system — metric (cm) or US/Imperial (in).
- Enter the number of points (n) — for example 5 for a pentagram.
- Enter the density / starriness (m) — the Schläfli number, with 2 ≤ m < n/2.
- Enter the outer radius (R) — the radius of the circle that passes through the star's points.
The calculator instantly returns the Schläfli symbol, the point angle, the inner radius, the inner angle, the edge length, the perimeter, the area, the drawn chord length, the diameter, and the number of vertices.
Unit Systems
Metric: enter dimensions in centimeters (cm); the perimeter is also shown in mm, m, and in.
US / Imperial: enter dimensions in inches (in); the perimeter is also shown in ft, yd, and cm.
FAQs
What is a star polygon?
A star polygon is a non-convex, self-intersecting, equilateral, and equiangular polygon. It is described by the Schläfli symbol {n/m}, where n is the number of points and m is the density.
What is the point angle of a five-pointed star?
For a pentagram {5/2}, the point (tip) angle is 180° × (5 − 2·2) / 5 = 36°. Each of the five points is a sharp 36° triangle tip.
Which values of m are allowed?
For a proper star polygon you need 2 ≤ m < n/2. For instance, with n = 7 you can use m = 2 or m = 3, giving the two distinct heptagrams {7/2} and {7/3}.
How many triangles are in a pentagram?
The most common answer is 35 triangles once you count every small, medium, and large triangle formed by the intersecting lines.
Is the Star of David a star polygon?
The hexagram {6/2} looks like a single six-pointed star but is technically a compound of two overlapping equilateral triangles, because 6 and 2 share a common factor.