If a square and a circle have the same perimeter. Which one has a larger area the square or the circle?

Question

A triangle, a square, a pentagon, a hexagon, an octagon and a circle all have an equal perimeter, which one has the smallest area?

Answer 1

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

Answer 2

If the side of the square is a, and the radius of the circle is r, then perimeter of the square is 4a and the circle is 2πr.

Given that 4a = 2πr → a = (πr)/2

Hence area of the square = a² or (π²r²)/4 and that of the circle - πr²

So, ratio of the areas = (π²r²)/4 : πr², Multipying by 4/πr² we get π : 4 or 22/7: 4 or multiplying by 7, we get 22 : 28

Therefore, area of the circle is more.