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A given 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 "squaring the circle". Due to the "irrational" and "transcendental" nature of number pi , squaring of circle is not possible 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 given. Is there any one who can support this claim?

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22 minutes ago, Ejup Dermaku said:

A given 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 "squaring the circle". Due to the "irrational" and "transcendental" nature of number pi , squaring of circle is not possible 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 given. Is there any one who can support this claim?

That is not what "squaring the circle" means. Given a circle of area 1, yes, there also does exist a square also of area 1. That is not a problem. The problem is that from a line segment of length equal to the radius (or equivalently the diameter) of such a circle, it is not possible only using ruler and compass to construct a line segment to make a side of a square of the same area as the circle.

The claim in your old book does not make immediate sense. It is true that if you are given a line segment of unit length, then you can quite obviously construct a square of unit area. But having been additionally given a circle of unit area would not be helpful in any way to do it.

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23 minutes ago, taeto said:

That is not what "squaring the circle" means. Given a circle of area 1, yes, there also does exist a square also of area 1. That is not a problem. The problem is that from a line segment of length equal to the radius (or equivalently the diameter) of such a circle, it is not possible only using ruler and compass to construct a line segment to make a side of a square of the same area as the circle.

The claim in your old book does not make immediate sense. It is true that if you are given a line segment of unit length, then you can quite obviously construct a square of unit area. But having been additionally given a circle of unit area would not be helpful in any way to do it.

Short and sweet.

Nothing more that needs adding here. +1

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