MAKING A SQUARE OUT OF A CICRLE
NOTE: all images are schematic and don't represent real values. They are just there to show the idea.

Problem: From a given circle create a square with the same area as the circle using classical tolls.

First let's recognise that the diameter of a circle and one side of a square with the same area always stand in a certain relationship to each other. Not depending on how large or small the circle is, since that would be just like looking at the same circle and square from a distance or up close:

pic1) same circle and square that has the same area from different distances.

Then we figure out a relationship:
Let's start backwards with a square with the side length 1cm. It's area would then be: 1 * 1 = 1 cm²
We can then figure out the circles diameter by: pr² = A where r is the radius, A is the area and p is a constant(approximately 3,1415). If we rearrange the formula we get that r equals:
r = sqrt(A/p) So since the area of the circle should be 1cm we can find out r by replacing A with 1:
r = sqrt(1/p) = 0,5641...cm
Since the diameter equals 2 radius' we can get the diameter that the circle with area 1cm must have by:
D = 2r = 2 * 0,5641... = 1,1283...cm
We then figure out the relationship between the side of the square and the diameter of the circle by dividing the square's side with the circles diameter:
1/1,1283= 0,6912... So the squares side is about 70% of the diameter of a circle with the same area, marked with a red line below on the circles diameter:

pic2)

The question now becomes; how do we separate this segment of the circle's diameter without measuring it up with a ruler? Well, let's first draw the diameter. We can split any circle in two using our compass: place the compass on the outline of the circle and draw another circle, then repeat this to draw a second circle; it doesn't matter where you draw the circles or how big they are as long as they overlap and aren't located on the same place on the outline. Then using our straight edge we can draw a line that goes through both of the places where the two newly drawn circles intersect:

pic3) The intersecting line is drawn in red, the two new circles drawn in green and the original circle drawn in white

The part of the red line that exists inside the original, white, circle; is the diameter since it goes straight through the circle.
Then we have to separate the 70% of the diameter that is needed for the circle. To do this we use the same method that we used for splitting the circle in a half to get it's diameter: You draw two arbitrarily large, intersecting circles whose centres are at the ends of the red diameter line; then you draw a line through the two intersecting points and this line will split the diameter in the middle(50%):

pic4)original circle: white, diameter of original circle: red intersecting circles: green, line splitting the diameter: blue.

Then you can get even closer to 70% by using the same method to split one of the 50% halfs you just got. To do this you draw one circle one whose centre is at the end of the diameter line and another circle whose centre is at the middle point of the diameter which you've just obtained. Make sure they intersect and draw a line through the intersecting points:
pic6) The right circle has it's centra at the middle of the diameter and the left one has it at the left end of the diameter.

You have now reached 75% measuring from the right to the left of the diameter(first splitting the diameter in 50% and then splitting one half into two making 25%, 50 + 25 = 75), so now you'll have to "go back" a bit to try to reach 70%. That means instead of splitting the diameter as to move the segment that will later become our squares side to the left, which we've done previously, we now have to move it to the right a bit. So now we draw the two intersecting circles whose centre's lies on the middle of the diameter and on the 75% mark of the diameter:

pic7)

Now we've come a bit closer to our 70%(we've reached 65,2%, counting from the right of the diameter to be exact). Using this method by splitting forward and backward you can slowly home in on 70%(you might have noticed what is wrong with this method now but I'll still wait with the explanation until the end of the page).

Now let's say that we've got the diameter split right where we wanted it, about 70%. How do we construct a square of this segment of the diameter? First we accommodate our compass so that it can draw a circle with a radius whose length corresponds to this 70% of the original circles diameter:

pic8

We draw this circle, note that it now has a diameter twice the value that the side of our square will have since our compass drew a circle that had the same radii length as our squares side will have. Then we draw the diameter of this new circle and then mark the middle point(50%) of it using the methods described above(pic 3-4). However now we wont just use the middle line to split the diameter further, but we'll elongate this line so that it also intersects our new circle at a point; 90^o above the middle point of the diameter we've draw:

pic9)The new circle with a diameter(red) that is twice the value that our square will have. The diameter is split in two by the blue line. Now, because of the diameter length being twice the value that the squares side's will have(I will from now on call this length 's') the radii is exactly that length and both the blue line segments on each side of the red line are one radii and both of the red segments on each side of the blue line are one radii, so the one of the blue segments and one of the red segments starts to look like a corner of the square.

Indeed it is the corner of our square! Now draw another circle with the diameter of 2's' whose centre is allocated on the tip of the blue line and then elongate the blue line further using our straight edge, it doesn't matter how far you elongate the blue line as long as it touches the outline of the circle we just drew:

pic10) The white circle is the first circle with diameter 2s that we drew and the purple circle is the second one we just recently drawn. I have extended the blue line from pic9 so that it intersects the purple circle and continues onward. The part of the blue line that exist inside the purple circle is, of course, the purple circles diameter.

We split the diameter of the purple circle(the part blue line inside the purple circle) by methods described in pic4. Then we elongate the line that splits the diameter so that it is at a 90^o angle from the middle of the diameter line as in pic9:

pic11)

Then we erase the circles:

pic12)

What have we got?! The ability to make a square, just use the straight edge to connect any of the two yellow-red tips and then erase all other stuff:

pic13)

And the area of this square is precisely the area of our original circle, since we created the sides out of the radii of another circle that had the diameter of 2's' i.e. twice the required length for the sides of a square with the same area as the original circle.

Why it doesn't work: Well first of all you need to do an infinite number of actions, when splitting the original circles diameter, since as you remember we didn't really want 70% of it but more of 6912...%. And this is an irrational number meaning that it contains an infinite amount of numbers(i.e. after 6912 there's an infinite amount of numbers). So you can say that it is infinitely accurate so you need to repeat the splitting action an infinite amount of times, and this is because of p that we used is an irrational number. Second of all, since you sometimes needed to split the diameter as to sometimes shorten and sometimes elongating the segment to get it to the wanted 70% by splitting it a bit more to the right and sometimes more to the left(pic6-7), someone else needed to have done it and written down the instructions before! As if that wasn't enough, this person needed to have used other tools then the classical ones to get the exact value for the percent of the original circles diameter that would become the squares side's.

However you might be able to remember enough splitting procedures as to be able to get it to a close enough value to the squares sides and impress some math teachers :).




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