Schläfli symbol calculator

Let's check out different star shapes!

If we start from the hexagon (the bestagon), we can build the hexagram. In particular, you can draw it from a pentagon.

🔎 You can draw a pentagram without separating the pen from the paper.

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.

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.

Star polygons are simple shapes, but somehow they keep appearing all around us.

Consider flags, for example.

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.

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). 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.

But it doesn’t teach. It looks like someone really put a lot of thought into such a simple geometric shape. 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.

Well, everything apart from occultism: we deal with geometry, not magic! 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!

Admire the proportions of these quantities:

Knowing this, we can almost understand why the pentagram got its reputation! Find out more about hexagon symmetries with our hexagon calculator.

We can calculate the area of a hexagram with the formula:

Increase the number of sides and get a heptagram, a seven-pointed star.

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. It’s not about rushing to the right answer. However, the number of stars rises quickly if you count the shapes: the US flag alone contributes with 50!

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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.

It highlights the rules in play and helps students see patterns.