Schläfli symbol calculator
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It analyzes what kind of problem has been inputted such as an equation, an integral, a system of inequalities, and generates the best way forward.
Whether a problem is typed, spoken in everyday language, or shared as a photo from a notebook, our AI is designed to help interpret and respond to it.
This flexibility matters, because sometimes learners don’t remember every symbol.
That’s a quiet but powerful difference. 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.
When we use this value in a trigonometric function like the tangent, we find tan60°=3. 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⋅2d=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.
Admire the proportions of these quantities:
al=ba=cb=φ
Knowing this, we can almost understand why the pentagram got its reputation! It doesn’t notice when a step is skipped or suggests a better path forward.
That’s where a math solver can offer more.
Symbolab’s AI Math Solver is built not to just compute, but to explain.
Not with flashy features or shortcuts—but by offering clear, step-by-step explanations designed to support learning.
Solutions are broken down in a structured way, helping users follow the logic behind each step — why it comes next and how it builds on the one before. 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.
Star Shape Calculator
Twinkle, twinkle, little star...
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.
What is the pentagram shape?
Let's meet the VIP (very important polygon) among the star polygons: the pentagram!
What is a pentagram?
The flag of Jordan depicts a heptagram; the shape was also part of the old Georgian coat of arms. Try to follow these tips:
- Identify the components of the shape: other stars or convex polygons.
- If there are separated elements, start from an intersection.
- Retrace an already drawn shape.
What is the perimeter of a pentagram with side 5?
This perimeter is .
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.
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:
P=2⋅n⋅b
However, there are many more "specific" formulas. Rather than relying on rigid, pre-programmed steps.
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. But you knew this already: everyone draws stars in their lives!
The pentagram got a bad reputation due to its association with evil: don't trust such voices and keep exploring this fascinating shape's geometry!
Here is a pentagram with all the important features highlighted!
You will meet these elements in all the other star polygons in this calculator: pay attention!
You can identify:
- a, the distance between two contiguous "points" of the star;
- b, the length of the first set of "rays" wounding around the polygon;
- c, the side of the original polygon; and
- l, the distance from two "points" of the stars belonging to the same side of the polygon.
Note that connecting the points of the star creates a new n-sided regular polygon with side a!
Some relationships between those quantities always hold (for every star polygon); others are specific for a given shape.
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.
It highlights the rules in play and helps students see patterns. 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.
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. 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.
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.