I am going to explain a small puzzle about a small mill wheel just so you can see that the big problem of the square of the circle, useful in our day to day, can be explained and taught in a simple way.
It is said that two honest Syrians put all their assets in common to buy a mill wheel. As they lived far from each other they agreed that the older man would keep the molar until using it his size would be reduced by half, at which point he would give it to the other.
The wheel was exactly 22 inches in diameter and a hole in the center of 3 1/7 for the handle, as shown in the drawing.
What size will the wheel be when it is returned to the second owner?
Our Syrian friends could draw the approximate number of square inches contained in a circle 22 inches in diameter. From here, deduct the number of inches contained in the hole 3 with 1/7. Then they will find out the approximate size of a circle that contains half of the square inches, which will be the size of the wheel when the first man has finished using it.
The only perfect method, however, is based on our demonstration that the area of circles can be computed from the squares of its diameter.
Knowing, thanks to Pythagoras, that a square inscribed in a circle will contain another circle that will measure just half the original, take the wheel and after drawing the lines from A to C and from B to D, make a square A, B , C, D; then draw a circle E, just inside that square.
However, we have said that the middle hole should be divided between the two mill owners. So we will draw a square within that circle, and within that square, we will make another circle that will measure half that of the first, F. And now we will launch the Pythagorean principle to add circles, and place the small circle in G, and the line from H to I will form the hypotenuse line of a right triangle, which gives us the diameter of a circle combining the area of E and the small circle that is half of F, which enlarges the circle E, so the dotted line shows a circle that contains exactly half of the mill wheel and that It has a diameter of 15 5/7 inches.