Squaring of the circle

Question

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:

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.

(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:

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.

Answer 1

It's not possible to construct the length of the circumference of the circle with compass and ruler.

Answer 2

"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.

Answer 3

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.