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?

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    $\begingroup$ What geometry exists to calculate area at all? For curves.... no, nothing exist that isn't essentially calculus. $\endgroup$ – fleablood Apr 11 '18 at 1:05
  • $\begingroup$ 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/… $\endgroup$ – John Wayland Bales Apr 11 '18 at 1:09
  • $\begingroup$ This question has a number of proofs including my own. math.stackexchange.com/questions/2593324/… $\endgroup$ – Rene Schipperus Apr 11 '18 at 1:10
  • $\begingroup$ 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. $\endgroup$ – 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.

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

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.


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