Schläfli symbol calculator

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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.

But most calculators don’t tell you why the answer is right or where things went wrong.

A basic calculator does what it’s told. When we use this value in a trigonometric function like the tangent, we find tan60°=3​. For example, you can insert the area, and we will tell you all the other lengths.

⚠️ The cotangent function in the area formulas of heptagrams and octagrams prevents the calculations from running in the inverse direction!

FAQs

What is a star-shaped polygon?

A star-shaped polygon (n-gram) is a non-convex regular polygon.

To the stars, with Omni Star Shape Calculator!

Star shapes: pointy geometry

Star polygons are the shiniest shapes in geometry. Let's see the ones for the pentagram:

​l=a⋅φb=φa​c=φb​​

φ is the golden ratio:

φ=21+5​​≃1.618

Only the pentagram among the star polygons has sides that relate to the ratio.

Of course, you can build irregular star polygons, but they would be impossible to calculate, and we are here for this!

We can uniquely identify every star polygon using the Schläfli symbol. How many triangles are in a pentagram shape?

We counted... The hexagram "rays" are nothing but equilateral triangles.

The pentagram is nothing but a five-pointed star shape. 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. The formula is:

​A=(4⋅2a​⋅d2−(2a​)2​)+(2⋅a2⋅cot(8π​))​

You can build star shapes for any regular polygon: the series continues with the enneagram, the decagram, etc.

And that’s okay

Symbolab’s AI Math Calculator is designed to meet students where they are and help them move forward.

Chooses a Structured Approach That Supports Understanding

Understanding the problem is just the beginning. 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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However, humanity covered it with a fair dose of mysticism.

if was in e.g. It breaks problems into step-by-step solutions, showing how each part contributes to the answer. 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.

schläfli symbol calculator

It’s familiar. 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.

That’s what Symbolab’s AI Math Solver is built for.

Instead of just giving a final answer, Symbolab’s AI Math Solver breaks problems down step-by-step. Well, everything apart from occultism: we deal with geometry, not magic! Do we need to mention the Star of David, a hexagram, or the shared symbolism associated with the pentagram from Hinduism to Neo-paganism?

You can also notice that the constructions of a star-shaped polygon create many triangles, the simplest bi-dimensional shape.

To make the game more entertaining, connect the outside points of the n-gram!

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. Here are the formulas for the side: don't worry if you see some 2​ factors, blame trigonometry instead!

​a=2⋅b⋅sin(135)c=b⋅2​d=b+c​

As for the heptagram, the fastest way to compute the area involves the separation of the octagram in an octagon with side a and eight triangles.