Math CentralQuandaries & Queries


Question from a student:

given a circle and square of same perimeter, why does the circle have a larger area

I'm looking for an intuitive explanation rather than a computational one. Thanks



I can give you a construction argument. I'm not sure how intuitive it is but it has no arithmetic calculations.

First I traced the circular base of a 3.36 L paint can and cut it out.


Next I wrapped a length of string around the can so that the string length was the length of the perimeter of the circle. I folded the string in half and then in half again to obtain the length of a side of a square with the same perimeter as the circle. I constructed a square with this side length and cut it out.



Finally I placed the square on top of the circle. Which has the larger area?

square and circle

Try it. You need a reasonably large circle to see the difference,

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