Squaring the Circle — Why It Is Impossible

What just happened?

You tried to "hit" the area of the circle with the area of the square. Equality holds for only one single value of the side: $a = r\sqrt{\pi}$. You can get arbitrarily close to it, but you will never hit it exactly by construction with compass and straightedge.

What is the problem?

"Squaring the circle" is one of the three great unsolvable construction problems of ancient Greece (alongside trisecting an arbitrary angle and doubling the cube): given a circle, construct with compass and straightedge a square of equal area. A circle of radius $r$ has area:

A square of the same area would have to have a side satisfying $a^2 = \pi r^2$, that is:

Why is it impossible?

With compass and straightedge you can construct only certain numbers — constructible numbers. These are numbers obtained from one by addition, subtraction, multiplication, division and square roots. All of them are algebraic (they are roots of polynomials with rational coefficients). However:

Since $\pi$ is transcendental, so is $\sqrt{\pi}$ — and therefore the side of the sought square $a = r\sqrt{\pi}$ does not belong to the constructible numbers. The construction is impossible.

The invariant — what decides the outcome:

Constructible with compass and straightedge are only numbers obtained by arithmetic operations and square roots (algebraic numbers of a certain type). The number $\pi$ (and hence $\sqrt{\pi}$) does not belong to them, because it is transcendental. This one property of the number $\pi$ settles the entire problem of over two thousand years.

Who settled it, and when?

For over 2000 years mathematicians tried in vain to "square the circle" — the phrase itself became a synonym for an impossible task. The solution came only in 1882, when Ferdinand von Lindemann proved that $\pi$ is a transcendental number. It is a beautiful example of how deep number theory settles a concrete geometric problem — showing not "we cannot do it yet", but "it has been proven impossible".

Why you need this for the exam:

It is a classic context for the concepts: the area of a circle $\pi r^2$, irrational and algebraic vs transcendental numbers, and the role of $\pi$. Remember the mechanism: the side of a square with the area of a circle is $r\sqrt{\pi}$, and $\pi$ is transcendental — which is why the construction is impossible.

Remember: squaring the circle is impossible with compass and straightedge, because the side of the sought square is $a = r\sqrt{\pi}$, and $\pi$ is a transcendental number (Lindemann, 1882) — it cannot be constructed.