If a square and a circle have the same perimeter. Which one will have the bigger area?

Mathematics

Your Answer

if the perimeter is fixed the circle will have the maximum area possible compared to any other shape including that of square.
We can check that as follows.
Let P be perimeter then, 
P = 2*π*radius {For Circle} = 4*side {For Square)
therefore
Circle Area = π*radius^(2) = π*(P/(2π))^(2) = P^2/(4π) = 0.07958*P^(2)
Square Area = side^(2) = (P/4)^(2) = P/16 = 0.0625*P^(2)
hence
Circle Area > Square Area
Let’s now find the area of circle (Ac)Ac) w.r.t its circumference/ perimeter (PP), where rr is the radius of the circle
P=2πrP=2πr
Therefore, r=P2πr=P2π
Ac=πr2Ac=πr2
Ac=π(P2π)2Ac=π(P2π)2
Ac=P24π(1)(1)Ac=P24π
Let’s now find the area of square (As)As) w.r.t its perimeter (PP), where ll is the side of the square
P=4lP=4l
l=P4l=P4
As=l2As=l2
As=P216(2)(2)As=P216
Let’s now find the area of equilateral triangle(Ae)Ae) w.r.t its perimeter (PP), where ss is the side of the triangle
P=3sP=3s
s=P3s=P3
Ae=3√4∗s2Ae=34∗s2

Ae=3√4∗P29Ae=34∗P29
Ae=P3√36Ae=P336
Ae=P123√(3)(3)Ae=P123
Comparing the denominators of equations 1, 2 and 3,
4π<16<123–√4π<16<123
As the above numbers are in the denominator,
Ac>As>AeAc>As>Ae

Thus, the circle contains he maximum area and the equilateral triangle contains the minimum area.

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