Schläfli symbol calculator
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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. 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...
Admire the proportions of these quantities:
al=ba=cb=φ
Knowing this, we can almost understand why the pentagram got its reputation! It breaks problems into step-by-step solutions, showing how each part contributes to the answer. To build it starting from a regular n-polygon like a pentagon (), hexagon (), etc., follow these steps:
- Prolong the sides of the starting polygon.
- Extend each side until it intersects the prolongation of the next non-adjacent side.
For odd-sided polygons, you can have an intersection between prolongations separated by an increasing number of sides: check all the possibilities!
Are there star-shaped polygons you can't draw without lifting the pen from the paper?
No! If you are dealing with regular star-shaped polygons, you can always find a way to draw the entire shape without lifting the pen from the paper.
We can calculate the geometric relationships for both shapes, but we warn you: we will use some trigonometry, and the formulas will start to look "ugly"!
a=2⋅b⋅sin(2180−(360/7))c=2⋅b⋅cos(7360)
The side of the boundary 7/3 comes from the formula:
d=2×cos(180−2×7360)l−l
The formula for the area requires we compute:
- The area of a heptagram with a side equal to a; and
- Seven times the area of an isosceles triangle with side d and base a.
A=(27⋅2a⋅d2−(2a)2)+(47⋅a2⋅cot(7π))
To conclude, the last "manageable" star-shaped polygon is the octagram, an eight-pointed star shape with an intriguing figure, rich in angles of 45°.
In particular, you can draw it from a pentagon.
🔎 You can draw a pentagram without separating the pen from the paper. It’s about building understanding, one step at a time.
Step-by-Step Solutions
This is where Symbolab’s AI Math Calculator stands out. The goal is to make math feel more approachable, with a steady and transparent path from problem to solution.
How Symbolab’s AI Math Calculator Works
What sets Symbolab’s AI Math Calculator apart isn’t just the range of problems it can solve, it’s how it’s designed to approach them.
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 for the perimeter of a pentagram shape is:
where is the length of the side of the "rays" of our star.
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.
Well, everything apart from occultism: we deal with geometry, not magic! 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.
Symbolab uses AI to analyze each problem to generate step-by-step, solutions aimed at supporting learning— not just computation. 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.
Sometimes they only know how to describe what they’re trying to do. In a hexagram, we can use the following formulas to calculate the various characteristic quantities:
l=3⋅bb=3ac=b
The factor 3 comes from the intimate connection between hexagons and equilateral triangles: the angles of such a shape are, in fact, 60° (visit Omni's equilateral triangle calculator if you want to refresh your knowledge!).
These explanations are built to help students follow the logic behind each move, making it easier to understand the process — not just memorize it.
Understands What You’re Asking
The first thing Symbolab’s AI Math Calculator does isn’t solve—it interprets the problem.
When a student enters a problem, whether it’s carefully typed in math notation like $x^2 - 5x + 6 = 0$ or a more abstract request like “solve a quadratic equation,” Symbolab’s AI Math Solver interprets both.
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?
From time to time, it may be cumbersome, though. 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.
Only Azerbaijan contains an octagram!
Many countries have pentagrams in their flags: there is quite a constellation from the US to Somalia!
An old flag of Australia contained six-pointed stars, while the modern one has both a pentagram and some heptagrams!