Schläfli symbol calculator
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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?
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. The hexagram "rays" are nothing but equilateral triangles. 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.
Symbolab uses AI to analyze each problem to generate step-by-step, solutions aimed at supporting learning— not just computation. 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. 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. Of the 196 country flags in the world, 40 of them contain stars. Well, everything apart from occultism: we deal with geometry, not magic! The AI is designed to respond in a clear, helpful, and reassuring way, encouraging you to take another look with confidence.
Practice Problems: Explore a wide range of examples, supported by hints designed to guide you towards the next step.
Quizzes: See how far you've come, with feedback that feels more like a mirror than a grade.
Because real understanding isn’t fast.
Most calculators are built to compute and move on. If a limit needs the chain rule, it explains why. 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. 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 more with the star shape calculator at Omni Calculator!
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It breaks problems into step-by-step solutions, showing how each part contributes to the answer. The formula for the perimeter of a pentagram shape is:where is the length of the side of the "rays" of our star. Admire the proportions of these quantities: