The answer to this question has been investigated by many mathematicians,
and it has been found that with a compass and straightedge construction
this is impossible.
Hippocrates found that there are certain regions with curved boundaries that are squarable (called lunes).
A construction similar to his for one squarable lune follows using GSP.
You start with an isosceles right triangle, and then semicircles are constructed on the three sides as shown in the following picture.
Consider the ratio of the areas of the semicircles on AB and AC. You get
Now since ABC is an isosceles triangle with AB the hypotenuse,
So, Area of the semicircle on AB has twice the area of the semicircle on AC.
One example of this follows.
Click here to view the GSP file that
Therefore the area of a quarter circle from AB is the same area as the semicircle on AC.
Now, if the area of the segment of the quarter circle is the commen area of the quarter circle on AB and the semicircle on AC
Therefore the area of the lune on AC and the area of triangle AGC are equal. Since AGC is an isosceles right triangle, its area is
Thus the area of the lune is the same as a square of side of length
An example of this follows
Click here to view the gsp file that
This, however is not a solution to squaring the circle.
Every lune cannot be squared. In fact this particular lune is one of 5 that can be squared.
The Quadratrix was invented by Hippias to trisect an angle.
The following is an example of a quadratrix. (Click here to view the GSP file that demonstrates this)
See how the three proportions are equal
The final proportion is (arc BED)/(arc ED)
While Hippias used the Quadratrix to trisect an angle, Pappus was able to demonstrate how it can be used to find a square equal in area to a given circle.
If it is assumed that point G (see Quadratrix above) can be found, then the following can be proved:
(arc BED)/AB = AB/AG
Pappus just assumed you would then be able to arrive at a square
equal in area to a given circle since 2 square lengths were used
to describe an arc length.
If you have a circle of radius r, then a line segment of length s can be constructed in which
(C/4)/r = r/s. This is the general formula for the above formula (arc BED)/AB = AB/AG.
arc BED was 1/4 of the original circle of radius AB. And the length s obtained was AG.
Therefore we know C/4 = (r)(r)/s
Let q = (r)(r)/s=C/4
The area of a circle is (pi)(r)(r), and the Circumference is 2(pi)(r), therefore the the area is (C/2)(r).
Burton, David M. Burton's History of Mathematics: An Introduction