elementary proof of circle area formula

Question

Can the area of a circle be derived without calculus or Archimedes approach? The area is given as$\pi R^2$, where $\pi$ is defined by circumference=$2\pi R$. It is easy to derive it as an integral or by using the limit as a sequence of polygons (Archimedes). Is there a more elementary geometry derivation?

Answer 1

An elementary proof cannot exist because $\pi$ is a transcendental number. It cannot be expressed as a fraction, nor even as the root of a polynomial with integer coefficients. It cannot be computed geometrically, with ruler and compass. To be expressed from integers using algebraic functions only, it requires an infinite process.

This problem is known as the impossible squaring of the circle.

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Note that if the formula for the circumference is taken for granted, the proof for the area is easy. But that just displaces the problem to that of the circumference.