The circumference of a certain circle and the perimeter of a certain sqarue are the same. Which has biger area

Question

Please help me. justify your answer in terms of the same variable and if you could explain how you came up with the answer, that would be super!

Answer 1

keep in mind c=p

For circle:

Circumfrence is 2 Radius x pi.....so Radius is C/2pi

Area is pi (c/2pi) ^2


For square

Perimeter = L x 4

so L = P/4

A = (P/4)^2

A from the first is pi(c/2pi) ^2 and area from second is A = (c/4)^2


So (C^2)/4pi to (C^2)/16 makes it

4pi : 1 ratio


Thus if circumfrence and perimeter are the same, then the circle is bigger

Answer 2

Let C be the circumference. The area of the corresponding circle is pi * r^2. Since C = 2 pi r, this area is pi * ( C / 2 pi )^2. The area of the corresponding square is ( C / 4 ) ^2. The ratio of the area of the circle to the area of the square is

pi * ( C / 2 pi )^2 / ( C / 4 )^2

which simplifies to 4 / pi > 1

So the circle always has the larger area than the square with the same perimeter as the circumference of the circle.