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From curiosity I read about enhancements of the squaring of the circle. Right now I do not see what is wrong with the following approach:

Archimedes devised that the surface of a circle equals the surface of a triangle which has one leg of the length of the radius and one leg has the length of the circumference of the circle, as shown in the following picture:

triangle with surface as a circle

Which just means: Surface of circle = 1/2 * radius * circumference

From this triangle, one can create a rectangle of equal surface by "cutting" the triangle into two halfes.

Creating rectangle

(Please excuse the loosey picture, it is just to visualize the idea)

And any rectangle can be tranformed into a square easily, as shown here:

No english source found

So, as far as I understood, the problem with squaring a circle is that PI is irrational.

But if PI is respected in constructing the square (as in the leg of the triangle), it seems legit at first glance (at least to me).

I know lots of very smart people tried this, so this approach is flawed and has already been thought.

Why is this construction not legal?

As I do not see any reason to use compass and ruler here, I want to express it in a more mathematical construction:

Assume a circle with radius r and circumfence of 2 π r and surface area = π * r^2

Create Archimedes orthogonal triangle with:

AO=r, BO=2 π r

Create rectangle from this triangle with:

AB = p = (1/2 * 2 π r) = π r

AC = q = r

Make a square of this:

h^2 = p * q = (π r) * (r) = π r^2

so

h = sqrt(π r^2)

With a surface of

sqrt(π r^2) ^2 = π r^2

So Surface square = Surface circle

q.e.d.

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    $\begingroup$ You said: "So, as far as I understood, the problem with squaring a circle is that PI is irrational.", but a number simply being irrational doesn't automatically make it impossible to be constructed. E.g., $\sqrt{2}$ can be constructed by creating a right triangle with the sides $1,1$. $\endgroup$ – user26486 Feb 18 '15 at 19:05
  • $\begingroup$ @user314 It's transcendent, so you can't get rid of it. But finding a square with equal surface area as a circle seems to be an easy problem. $\endgroup$ – Mare Infinitus Feb 18 '15 at 23:15
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It's not possible to construct the length of the circumference of the circle with compass and ruler.

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  • $\begingroup$ You can do it as exact as you want with PI with as many decimal places as you want. At least I do not think this answers my question. I understand that PI will never end, so the length will never be exact, but you can calculate as exact as you can. And: You would never know the exact circumference more exact $\endgroup$ – Mare Infinitus Feb 18 '15 at 19:03
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    $\begingroup$ The original problem is about squaring the circle exactly, which is not possible because you can't construct $\pi$ exactly with finitely many steps. $\endgroup$ – Ward Beullens Feb 18 '15 at 19:05
  • $\begingroup$ @MareInfinitus The question of squaring the circle is the question of constructing a square with exactly the area of the circle. You cannot take approximation, otherwise it's not a squarec circle. $\endgroup$ – Wojowu Feb 18 '15 at 19:05
  • $\begingroup$ This is what I understand as the problem here: You cannot create a square if you do not know the exact circumference. $\endgroup$ – Mare Infinitus Feb 18 '15 at 19:07
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    $\begingroup$ @MareInfinitus All the steps would've been legitimate and this would be a complete proof that you can square the circle if there was an actual way of creating a segment (the other leg of the triangle) of the length of the circumference of the circle. $\endgroup$ – user26486 Feb 18 '15 at 19:13
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"Unrolling" a circle as well as getting a line segment of non-unit length are not allowed operations. If they were, you could allow "get a square of area $\pi$" as well.

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  • $\begingroup$ No problem. Just take a square with side-length of sqrt(pi). $\endgroup$ – Mare Infinitus Feb 18 '15 at 19:48
  • $\begingroup$ Why not? Please have a look at the more mathematical construction. Why is this not allowed? $\endgroup$ – Mare Infinitus Feb 18 '15 at 22:44
  • $\begingroup$ It makes it a completely uninteresting problem if among the tools there is a magical one that gives the solution instantly. $\endgroup$ – Yves Daoust Feb 19 '15 at 8:54
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The classical problem called "squaring the circle" means the problem of doing it with compass and ruler. If you want to use other mechanisms (such as rolling a disk) you are solving a different problem. As you observed, the "unrolling" technique was thought of many centuries in the past; making a square with the same area as the resulting triangle was a trivial problem to ancient Greek geometers and hardly worth mentioning.

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  • $\begingroup$ But making a square with the same area as a circle is the problem? What is the difference with "compass and ruler" and the problem solved here? $\endgroup$ – Mare Infinitus Feb 18 '15 at 22:48
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$π$ is a transcendental number and could not be constructed with compass and straightedge. However, it is feasible to construct a square with an area quite close to that of a given circle. If a rational number is used as an approximation of π, then squaring the circle becomes a possibility. If some $π$-approximation is used to draw a line equal to π (appr.) so circle could be squared with area quite close to the circle itself.

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  • $\begingroup$ You make a good point. $\endgroup$ – Max0815 Mar 25 '19 at 15:25

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