How to Determine a Square and Circle of Equal Area

The relationship is that the perimeter of the square is equal to the circumference of the circle multiplied by 1.13. Thus, p = 1.13 c. Here's how that's derived: the circle's area (πr²) is defined as being equal to the square's area (4s), where r is the circle's radius, and s is the square's side. So πr² = s², making s equal to r√π. The square's perimeter is 4s, or 4r√π. You would have to multiply the circumference (2πr) by (2/√π), or 1.13, to get the square's perimeter (4r√π). So the relationship of circumference to perimeter is p = 1.13 c. This is true only when the two areas are equal.

A square with an area of 4 cm² has sides of 2 cm each and a diagonal of 2√2 cm. A circle whose radius is 2√2 cm has an area of π(2√2)² = 8π = 25.13 cm².

If the square's area is A, each side is √A. So the square's perimeter is 4√A, and so is the circle's circumference. Then the diameter of the circle is its circumference divided by pi, or (4√A) / π, meaning its radius is (2√A) / π. Then the circle's area is πr², or π(2√A / π)² = 4A / π² = (0.405)(A), or (.4A).