SOLUTION: What is the ratio of the square's area to the circle's area if the square and circle have equal perimeter?

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Question 237229: What is the ratio of the square's area to the circle's area if the square and circle have equal perimeter?
Answer by rapaljer(4551) About Me  (Show Source):
You can put this solution on YOUR website!
Let r= radius of the circle, and let x = side of the square.

Perimeter of circle = 2%2Api%2Ar.
Perimeter of rectangle = 4x

If the perimeters are equal, then
2%2Api%2Ar=4x

You need to find the ratio of the area of the square to the area of the circle.
Area of circle = pi%2Ar%5E2
Area of square =x%5E2

Ratio of area of square to area of circle = %28x%5E2%29%2F%28pi%2Ar%5E2%29

Before proceeding, just for fun (and it will be helpful later!!), find the ratio of x%2Fr using the equation above 4x=2%2Api%2Ar. Divide both sides by 2%2Api%2Ar:

+%284x%29%2F%282%2Api%2Ar%29=1
Divide out the 2 factor:
%282x%29%2F%28pi%2Ar%29=+1

Divide both sides by 2, and multiply both sides by pi.
x%2Fr=+pi%2F2.

NOW:

RATIO of AREAS= %28Area+of+Square%29%2F%28Area+of+Circle%29
=%28x%5E2%29%2F%28pi%2Ar%5E2%29=%281%2Fpi%29%2A%28x%2Fr%29%5E2
=%281%2Fpi%29%2A%28pi%2F2%29%5E2
=%281%2Fpi%29%2A%28pi%5E2%2F2%5E2%29
=pi%2F4

This is quite an interesting problem! You may want to get a second opinion on it!!

Dr. Robert J. Rapalje