Schläfli symbol calculator

By definition, they are:

• Non-convex;
• Self-intersecting;
• Equilateral; and
• Equiangular polygons.

Let's look at all of these features one by one!

• Non-convex means that you can draw a line between two points inside the star-shaped polygon that passes outside its perimeter.
• Self-intersecting means that you can build all the different star shapes starting from a regular polygon and prolonging its vertexes until the extensions cross each other.
• Equilateral signifies that all the outer sides of the star-shaped polygon have the same length (latus is the Latin for side).
• Equiangular means that all the angles lying in the same "region" are equal.

The last two attributes of the star polygons automatically qualify them as regular polygons.

Let's check the general ones first:

• The "side" of a star is given by the sum l=c+2⋅b; and
• You can calculate the perimeter of the first boundary of an n-pointed star with the formula:

However, there are many more "specific" formulas. This symbol consists of a pair of numbers in the form n/m, where:

• n is the number of corners/sides of the polygon; and
• m is its starriness or how many distinct boundaries you can identify around the center.

Let's see how to build a star polygon, and then this notation will be clearer!

This five-pointed star polygon has the Schläfli symbol 5/2, and it is so important we will deal with it in the next section.

Let's meet the VIP (very important polygon) among the star polygons: the pentagram!

What is a pentagram?

Let's see the ones for the pentagram:

φ is the golden ratio:

Only the pentagram among the star polygons has sides that relate to the ratio. It’s slow, layered, sometimes stubborn, and it needs space to grow.

Why Choose Symbolab Over Other Math Calculators?

Plenty of calculators can compute an answer.

For the perimeter, use the general formula above, substituting n=5.

Over time, this makes math feel less like guessing—and more like following a path you understand. From time to time, it may be cumbersome, though. 😉

Burundi's and Israel's flags contain hexagrams. Its features were built with the student experience in mind: the stress of not knowing where to start, the frustration of getting stuck halfway, the quiet relief when something finally makes sense.

From the way it breaks down complex steps to how it gently explains why each one matters, Symbolab’s AI Math Calculator helps make math problems feel more approachable and manageable.

The heptagram has a particularity: it is the first star-shaped polygon for which we can naturally find a third boundary.

The reason for this peculiarity is that two sides of a heptagon separated by two others can finally meet if prolonged. degrees mode (or any other) //////////////////////////////////////////////////////////////// function Text() {class Text {print (t, str, scale = 1, italic = 0, kerning = 1) {let fc = t._fullCircle;t.radians();let pos = [t.x(), t.y()], h = t.h(), o = pos;str.split('').map(c => {const i = c.charCodeAt(0) - 32;if (i < 0 ) {pos = o = this.rotAdd([0, 48*scale], o, h);} else if (i > 96 ) {pos = this.rotAdd([16*scale, 0], o, h);} else {const d = dat[i], lt = d[0]*scale, rt = d[1]*scale, paths = d[2];paths.map( p => {t.up();p.map( s=> {t.goto(this.rotAdd([(s[0]-s[1]*italic)*scale - lt, s[1]*scale], pos, h));t.down();});});pos = this.rotAdd([(rt - lt)*kerning, 0], pos, h);}});t._fullCircle = fc;}rotAdd (a, b, h) {return [Math.cos(h)*a[0] - Math.sin(h)*a[1] + b[0], Math.cos(h)*a[1] + Math.split('>').map(r=> { return [r.charCodeAt(0)-106,r.charCodeAt(1)-106, r.substr(2).split('<').map(a => {const ret = []; for (let i=0; i<a.length; i+=2) {ret.push(a.substr(i, 2).split('').map(b => b.charCodeAt(0)-106));} return ret; })]; });return new Text();}

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