If a square and a circle have the same perimeter why does the square have a bigger area
Answers
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
OrIf 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.
Answer:
Step-by-step explanation:
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
OrIf 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.
Hope this answer helps you ☺☺........