Geometry Question?

Question

The circumference of a circle and the perimeter of a square are the same.

Which has a bigger area-the circle or the square?

WORK MUST BE SHOWN TO EXPLAIN!

PLZ HELP!!

Answer 1

Let C be the circumference of the circle and the perimeter of the square. Also let w be the width of the square and r be the radius of the circle. Then we have:

C=4w, so w=C/4, and
C=2πr, so r=C/(2π)

The area of the square is then:

w² = (C/4)² = 1/16 C²

And the area of the circle is:

πr² = π(C/(2π))² = πC²/(4π²) = 1/(4π) C²

Since 1/(4π) > 1/16, we have that the area of the circle is greater than the area of the square.

Answer 2

Circumference of circle = 2πr
Perimeter of square = 4(Length) = 4l
As 2πr = 4l , r = (2l/π)
Area of square = l²
Area of Circle = πr² = π(2l/π)² = π(4l²/π²) = (4/π)l²
As 4 is greater than π, (4/π) will be greater than 1. Hence Area of circle is greater than area of square.