What is the ratio of the areas of a circle and a square if their perimeters are the same?

Answer 1

x = length of one side of square and r = radius of the circle therefore 4x = 2r * Pi therefore the radius of the circle is 2x/Pi therefore in terms of area x * x = Pi * ( 2x/Pi) * (2x/Pi) which gives a ratio of 1 : 4/Pi square area to circle area or 1 : Pi/4 circle to square x = length of one side of square and r = radius of the circle therefore 4x = 2r * Pi therefore the radius of the circle is 2x/Pi therefore in terms of area x * x = Pi * ( 2x/Pi) * (2x/Pi) which gives a ratio of 1 : 4/Pi square area to circle area or 1 : Pi/4 circle to square
No, not as a universal rule. To illustrate, think of a piece of string that is 12 feet long. That piece of string can go around the perimeter of a square that is 3 feet on each side (i.e. adding up the 4 sides, each 3 feet long, would yield a square that has a 12 foot perimeter). The area of that same square would be calculated as 3 times 3 which equals 9 square feet. Now picture that same string going around a rectangle that is 2 feet wide by 4 feet long. This is thus a shape that also happens to have a 12 foot perimeter. But the area for this shape would be 2 times 4 which equals 8 square feet. Thus two different shapes with identical perimeters do not have to have the same area. This simple illustration with two common shapes (a square and a rectangle) that have identical perimeters but different areas can be extended to the odd shapes. Having the same perimeter does not lead to the conclusion that the shapes then have the same area. Hope this helps, I had to think about it myself!