Hamid Geometry Problem

Geometry Level pending

What is the relationship between the area of an equilateral triangle and an n n -sided regular polygon having the same perimeter.

Hint: A triangular has 3 sides of length x x : perimeter P = 3 x P=3x and area S △ = 1 2 x 2 sin ⁡ 6 0 ∘ S_\triangle =\frac 12 x^2 \sin 60^\circ . A regular polygon has n n sides of length b b : P = n b P=nb ; S p = F ( b ) S_p =F(b) . If 3 x = n b 3x = nb , then obtain S p = H ( x ) S_p = H(x) ; H ( x ) = ? H(x)=?

3/2 * n * x^2 * cot(180/n) 3/2 * n * x^2 * tan(180/n) 9/4 * n * x^2 * cot(180/n) 9/4 * n * x^2 * tan(180/n)

Hamid Javaheri by hamid Javaheri , 31, Iran

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1 solution

Hamid Javaheri
Jun 27, 2017

For a triangular we have : P=3x; S = 1/4 * x * sin(60) for a n-polygon we have P=nb The area of n-regular polygon is the product of "n" number of Equilateral triangle that has sides "b","r","r"; By obtaining the sides we have : S = n(1/2 * b * r * sin (90 - 180/n)) r=b/(2*sin(180/n)); b = 3x / n;

Then S = 9/4 * n * x^2 * cot(180/n)

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