When does a Triangle have Maximum Area?

Geometry Level 3

Find maximum area of the triangle whose semiperimeter is S

S 2 3 3 \dfrac{S^{2}}{3\sqrt{3}} S 2 3 \dfrac{S^{2}}{3} S 2 4 \dfrac{S^{2}}{4} S 2 2 3 \dfrac{S^{2}}{2\sqrt{3}}

Aditya Chauhan by Aditya Chauhan , 20, India

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1 solution

Edwin Gray
Jun 21, 2018

I believe that it is well known that of all triangles with the same perimeter, the one of largest area is the equilateral triangle. If its side is a, then S = 3a/2. The area of an equilateral triangle of side a is (sqrt(3)/4) a^2 or (sqrt(3)/4) (2S/3)^2 =sqrt(3) s^2 = sqrt(3) S^2/9 = S^2(3sqrt(3)). Ed Gray

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