elementary proof of circle area formula

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?

• What geometry exists to calculate area at all? For curves.... no, nothing exist that isn't essentially calculus. – fleablood Apr 11 '18 at 1:05
• Here is an intuitive approach to motivate the conclusion that the area of a circle equals half the product of the radius and circumference: quora.com/… – John Wayland Bales Apr 11 '18 at 1:09
• This question has a number of proofs including my own. math.stackexchange.com/questions/2593324/… – Rene Schipperus Apr 11 '18 at 1:10
• I'm also the type of person who likes to know more of what is going on in a problem so I upvoted this question. – Timothy Aug 29 at 20:43

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.