Squaring the Circle when Circle area is 1

[Ejup Dermaku]

Circle with area A = 1 has a radius = 1/sqrt(pi). In this case there exist a square with the sides of length = 1 which has an area equal to 1. This problem is referred as a "problem of squaring the circle". Because of "irrational" and "transcendental" nature of number pi , it is well known that squaring of circle is impossible to be constructed only by ruler and compass. However I've read in an old mathematical book, that a construction is possible only in case when circle area A=1, without any further explanation. Is there any one who can support this claim?

 

[Dr.Peterson]

The trouble there is that you can't construct a circle with area 1! Squaring a circle refers to constructing a square with the same area as a given circle; but given such a circle, you can't determine its area to be exactly 1, and given the area, you can't construct the circle. So constructing the square is moot.

So I'd say they're wrong.

 

[Mr. Bland]

The reason it doesn't work is entirely because [MATH]\pi[/MATH] is transcendental: you can't make it by manipulating a finite number of integers. Since the squaring-the-circle problem is fundamentally about manipulating integers (they can be used to build fractions), it's impossible to bridge the gap. This remains the case even if the circle has an area of 1.