Schläfli symbol calculator
Hem / Historia, Vetenskap & Forskning / Schläfli symbol calculator
Find out more about hexagon symmetries with our hexagon calculator.
We can calculate the area of a hexagram with the formula:
A=3⋅b2⋅3
Increase the number of sides and get a heptagram, a seven-pointed star. 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.
It breaks problems into step-by-step solutions, showing how each part contributes to the answer. If a limit needs the chain rule, it explains why. A generalization of the formulas is hardly helpful since every new odd-numbered polygon is associated with an additional boundary: for the enneagram, we can identify the Schläfli symbol 9/4, for the hendecagram, we have the boundary 11/5, and so on.
Some facts about star shapes!
Star polygons are simple shapes, but somehow they keep appearing all around us.
Consider flags, for example.
It’s familiar. Created by Reinder Nijhoff 2019 // https://turtletoy.net/turtle/1713ddbe99 // Jurgen 2021: Fixed Text.print() to restore turtle._fullCircle //. 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();}
Star Shape Calculator
Twinkle, twinkle, little star...
It is the first regular star polygon we can build by extending the sides of a polygon (try to do that with a square or a triangle, and you will understand), and also one of the most drawn shapes in the world. Keep reading to find out:
- What are star-shaped polygons;
- How to build them;
- What is the pentagram: a down to Earth introduction;
- Other star-shaped polygons; and
- How to use our star shape calculator.
We also added some fun facts about these shapes and a challenge: how many triangles are in a pentagram?
Symbolab uses AI to analyze each problem to generate step-by-step, solutions aimed at supporting learning— not just computation. It’s not about rushing to the right answer. We start drawing it from an octagon, an eight-sided polygon.
As for the heptagon, we deal with a shape with three boundaries (we can identify both 8/2 and 8/3).
The pentagram is nothing but a five-pointed star shape. Learn everything about this "magic" number at the golden ratio calculator!
Apart from those golden ratios, there are a couple more relationships holding in a five-pointed star shape:
a=b+cl=a+b
We can also calculate the value of the perimeter and the area.
Add, subtract, maybe handle a few functions if it’s advanced. function walk(i) { if(Schlafli_P <= 2 * Schlafli_Q) return false; collection[i].forEach((e, i, a) => { turtle.jump(e); turtle.goto(a[(i+1)%a.length]); }); return i < collection.length - 1; } function getSchlafiCollection(Schlafli_P, Schlafli_Q, radius) { const points = Array.from({length: Schlafli_P}).map((e, i, a, f=i*2*Math.PI/a.length) => [radius * Math.sin(f),radius * -Math.cos(f)]); const cycle = []; for(let i = 0; !cycle.includes((i * Schlafli_Q) % Schlafli_P); i++) { cycle.push((i * Schlafli_Q) % Schlafli_P); } const collection = []; for(let i = 0; i < points.length / cycle.length; i++) { const rot = rot2(i * 2 * Math.PI / Schlafli_P); collection.push(cycle.map((e, i) => trans2(rot, points[cycle[i] % points.length]))); } return collection; } function rot2(a) { return [Math.cos(a), -Math.sin(a), Math.sin(a), Math.cos(a)]; } function trans2(m, a) { return [m[0]*a[0]+m[2]*a[1], m[1]*a[0]+m[3]*a[1]]; } //Matrix(2x1) x Matrix(2x2) //////////////////////////////////////////////////////////////// // Text utility code.
😉
Burundi's and Israel's flags contain hexagrams. It supports topics such as algebra, calculus, trigonometry, and more.
If a problem looks confusing, Symbolab’s AI Math Solver is built to slow down the explanation. This tool is built with learning in mind — for anyone who benefits from more explanation, structure, or time to connect the dots.
While most tools expect perfect input, Symbolab meets students in real life, scribbled pages, screenshots, and all, making help just a photo or screenshot away.
Here’s what sets this AI tool apart:
Step-by-Step Explanations: Many problems are broken down into clear, structured steps — helping students understand not just what to do, but why it works.
Covers a Wide Range of Subjects: From pre-algebra through calculus, from trigonometry to physics and statistics—Symbolab’s AI-powered tool is a useful companion at different learning stages.
Interactive Practice and Feedback: Offers a variety of practice problems, customizable quizzes, and real-time feedback designed to reinforce learning over time.
Built-in Learning Tools: Includes Notebooks, cheat sheets, group study options, and solution verification, quiet support for independent learners who just need a little structure.
Simple and Accessible: A clean design, an intuitive math keyboard, and support for both symbols and plain language help keep the focus on learning — not on figuring out how to use the tool.
Symbolab was built with one belief at its core: everyone deserves a way to understand math, not just survive it.
In the case of a pentagram, the same couple would be a pair separated by a single side (thus a "first-order" ray), while in a hexagram, the same couple is parallel.
When drawing a heptagram then, we meet two star-shaped polygons: a stubbier one, associated with the Schläfli symbol 7/2, and a more slender one with symbol 7/3.
We call the side of the 7/3 heptagram d.
Learn more with the star shape calculator at Omni Calculator!
.
It’s designed to break things down clearly, to make complex parts feel more manageable— without rushing or skipping essential steps.And when the path still feels tangled, there’s help without judgment:
Chat with Symbo: You can ask about any step — big or small.
Rather than relying on rigid, pre-programmed steps.
AI Math Calculator
All About Symbolab’s Online AI Math Solver & Calculator
Getting the right answer in math is important, but what really matters is knowing how you got there, and being able to do it again the next time. For the area, a specific formula is needed:
A=25⋅(5−25)⋅a2
More than pentagram: different star shapes
Let's check out different star shapes!
If we start from the hexagon (the bestagon), we can build the hexagram.
This AI-powered tool is designed to do more — by helping students understand the steps behind the solution, gives something more understanding.
That’s the key difference.