eli5: What does it mean that you can’t “square a circle”? Couldn’t you just take a circle with diameter 2, and then a 2×2 square ?
In: Mathematics
A circle with diameter 2 has an area of pi. A 2×2 square has an area of 4.
“Squaring the circle” means, using only a compass and a straightedge, constructing a circle and a square with equal area. (Not a square inscribed within or circumscribed around a circle.)
It turns out there is no way to manipulate angles, rays, arcs, and segments using compass techniques to convert between the side length and radius length that you need.
There are a variety of other tasks, like “Take one triangle and construct another triangle with the same area” or “Take a triangle, and divide it into two triangles with equal areas” that are possible with only a compass and straightedge. But “squaring the circle” is not one of them.
The square will have a area of 2 * 2= 4 but the circle area is pi * 2^2 /4 = pi ~3.1415…
The problem of squaring the circle is to make a square with the same area as the circle or vise versa. The only allowed tools are a compass and a straightedge and a finite number of steps.
The square need to have sides of sqrt (pi) ~1.772… so you need to get exactly that length from the circle with just a compass and a straightedge. This have been frooven to be impossible in 1882. PI is what is called a transcendental number, that is not a root of a polynomial with rational cooeficents. It was know before that if pi was a transcendental number the problem would be impossible to solve, it was the proof that pi was transcendental that was from 1882.
You can create a approximation with the tools, the more steps you use the closer you get but to get the exact correct value you need a infinite number of steps.
You example is creating a square with the same side as the diameter of the circle and how to do that have been known since antiquity. Here are one method [https://mathbitsnotebook.com/Geometry/Constructions/CCconstructionSquare.html](https://mathbitsnotebook.com/Geometry/Constructions/CCconstructionSquare.html)
The problem is to square the circle using *only a compass and a straightedge.* It cannot be done in a finite number of steps, which surprises us. It seems like we *should* be able to do it, but we can’t because of the fundamental nature of circles.
Squaring the circle was a problem the Greeks couldn’t solve. The question is how do you make a square and circle with the same area. Obviously you take a circle with an area of radius 1, and a square will with side length sqrt(π) will have the same area.
The problem is you can’t do that with just geometry, and that’s the problem the Greeks couldn’t solve. It becomes rather elementary once you learn algebra, which the Greeks didn’t have.
A circle with diameter 2 has an area of pi.
To make a square that has an area of pi, you’ll need the sides to be the square root of pi, and since pi is a seemingly endless number, so is its square root.
Good explanations have already been given, I just want to introduce you to the other two impossible problems from the ancient Greeks: [doubling the cube](https://en.wikipedia.org/wiki/Doubling_the_cube) (given a cube, make another cube of double the volume), and [trisecting an angle](https://en.wikipedia.org/wiki/Angle_trisection) (given any angle, divide it in three equal angles.) All of them equally impossible, but squaring the circle has a lot more popular appeal.
Cut circle into even quarters
Move each quarter to the diagonal opposite side
Boom
Now it’s a square
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