Name: Margaret

Who is asking: Student
Level: Secondary

Question:
I have the following question that I cannot answer on my own (I don't know where to begin): Is it possible construct a square whose area will equal the area of a given circle? Please Explain why or why not.

Margaret

This is a very old classical problem that interested the Greek mathematicians about 200 BC. In the late 1800's it was shown that such a construction is impossible. The reason that it is impossible has to do with the number pi. If you have a circle of radius 1 then its area is pi. If you can construct a square with area pi then its side length would be the square root of pi, that is you would be able to construct a line segment of length the square root of pi. But pi is transendental which implies that neither pi nor its square root can be obtained by a ruler and compass construction. If you are interested in a proof you can find one in the Geometry Forum.