This is what keeps me up at 2 o'clock in the morning

I want to write the formula that gives the area of a regular polygon with base bb and no. of angles nn. See the figure I made [made on Paint Mircosoft]!

We can compose any polygon made of triangles, which always have 2 equal sides (Hexagon has 60-60-60 triangles, but it fits here), and the length of the triangle ll perpendicular to the base.

We can start to say that the area of the polygon is:

Spolygon=b×l2×n S_{polygon} = \frac{ b \times l }{ 2 } \times n  "n" is the no. of triangles in the polygon \text{ "n" is the no. of triangles in the polygon}

We can express ll with trigonometry:

  • tanθ×b2=l \frac{ \tan \theta \times b }{ 2 } = l

We replace this at the first formula:

  • Spolygon=b×tanθ×b22×n S_{polygon} = \frac{ b \times \frac{ \tan \theta \times b }{ 2 } }{ 2 } \times n
  • Spolygon=b2×tanθ×n4 S_{polygon} = \frac{ b^2 \times \tan \theta \times n }{ 4 }

How can we define θ\theta?

Remember that the sides of the regular polygon can be found using the formula angle=(n2)×180n angle = \frac{(n - 2) \times 180 }{n} , and for this case, it should be θ=(n2)×180n×2 \theta = \frac{(n - 2) \times 180 }{n \times 2} , and, now we have the full formula:

Spolygon=b2×tan((n2)×90n)×n4 \boxed{ S_{polygon} = \frac{ b^2 \times \tan (\frac{(n - 2) \times 90 }{n}) \times n }{ 4 } }

Please correct me for any mistake, typo or anything else against the guidelines. Ciao!

Note by Nikolas Кraj
1 year, 5 months ago

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One thing I might add during my work, can the Brilliant team enable those features of editing with LaTeX LaTeX in creating notes/discussions, like those in creating problems or posting solutions to a problem?

Nikolas Кraj - 1 year, 5 months ago

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you can actually simplify the expression tan(((n-2)*90)/n) to tan(90-180/n) which reduces to (tan(180/n)^-1

Chitrank Gupta - 1 year, 4 months ago

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@Nikolas Kraj, you wrote spelling of "polygon" as "poligon"

Prabhnoor Singh - 1 year, 3 months ago

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Thanks, I edited it. Please next time give a hint of where I made a typo (f.ex. you spelled side wrong at "... the siles in the polygon ... ".).

Nikolas Кraj - 1 year, 3 months ago

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Help at

Joel Chamberlain - 1 year, 3 months ago

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What is best for high quality party supplies as i am arranging a party and don't know from where I can buy.

Aesh A - 1 year, 4 months ago

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