Schläfli symbol calculator
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However, the number of stars rises quickly if you count the shapes: the US flag alone contributes with 50! The designers of the national colors couldn't make up their minds!
Our favorite starred flag is the flag of Nepal: the only non-rectangular flag in the world also has a beautiful 12-pointed star which really makes it stand out!
Star-shaped polygons appear widely through sacred geometry.
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! 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.
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. 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!).
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. It’s familiar. 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.
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!
What matters next is how to work through it in a way that builds clarity and confidence.
Symbolab’s AI Math Calculator doesn’t just display a final answer. Find out more about hexagon symmetries with our hexagon calculator.
We can calculate the area of a hexagram with the formula:
A=3⋅b2⋅3
Increase the number of sides and get a heptagram, a seven-pointed star.
It’s not about rushing to the right answer. The pentagram is nothing but a five-pointed star shape. The flag of Jordan depicts a heptagram; the shape was also part of the old Georgian coat of arms. Learn more with the star shape calculator at Omni Calculator!
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Since there are five rays, with two sides each, to calculate the perimeter of the pentagram, it suffices to count ten times the length of .