An Exact “Construction”

We have shown that it is impossible to square a circle using a straightedge and compass, and we have looked at several approximate solutions to the problem of square the circle. Although it is not possible to square the circle using a straightedge and compass, it is possible to square the circle using other methods. This method of squaring the circle involves rolling a circle for half a turn. “Rolling a circle” is not something that can be done using a compass and straightedge.

Suppose a circle having radius of length one is rolled half a turn along a straight horizontal line. Let the circle begin the roll at point A and end at point B. Since the radius of the circle is one, the circumference is 2p. Since the circle rolls from A to B for half a turn, AB = p.

Now extend the segment AB to the point C so that AC = AB + 1 = p + 1. Construct a semicircle with diameter AC. Now draw a line through B perpendicular to AB, and let its point of intersection with the semicircle be D. We know AB = p and BC = 1, so since AB*BC = BD² , we have that BD = Öp. (Click here for the proof). Thus, if we construct a square having BD as one of its sides, the area of the square is exactly equal to the area of the circle.

Click here (and scroll down) to see the diagram that accompanies this construction.

This proof is from the following cut-the-knot web page: http://www.cut-the-knot.com/impossible/sq_circle.html.