I was looking through a book talking about the history of the number pi and it was talking for a few pages about the search for a way to find a circle and square of equal area. It talked about the Egyptians and their method and went on to the Greek ideas but never exactly explained what the problem is and why there isn't a solution. I ran out of time looking through the book, so maybe I missed this later on, but could someone explain a little about this?

The problem is that pi is transcendental. No matter how big the circle, and no matter how big the square, you could never have a circle with integral radius and a square with integral side lengths with equal area.

 

That's all there is to it, really. You could also ask how many squares it would take to make a circle (like in

 

 

), and then the answer would be 'infinite', as you can see by the picture.