mathman

The ratio of the circumference of a circle to its diameter (2r) is given by $\pi$ (definition). The area of a circle is $\pi r^2$. Can the area be derived without calculus?

Maschke

Archimedes did so in 260BC.

https://en.wikipedia.org/wiki/Area_o...es's_proof

http://www.ams.org/publicoutreach/fe...umn/fc-2012-02

romsek

Are limits calculus?

The limit of the area of a regular n-gon with "radius", (i.e. center to vertex length), $r$ has area

$A_n = n \sin^2\left(\dfrac \pi n\right) \cot\left(\dfrac \pi n\right)$

and

$\lim \limits_{n\to \infty}~A_n = \pi r^2$

SDK

It seems to me that Archimedes proof (as presented in your link) explicitly uses calculus. The fact that he didn't realize the thing he was computing was a limit is certain. However, that is exactly what is being described in the 2nd link. In fact, I would say he proved it using the squeeze theorem.

Though I admit I don't know any proof which doesn't require some kind of limit so I would be really interested if someone presented something.

Maschke

Well sure, and Eudoxus invented integration. But calculus typically means the organized and formalized body of theory and technique that started with Newton and Leibniz. So now it comes down to what we mean by the word calculus. The underlying ideas are very old.

SDK

I can agree that the important thing becomes "what does without calculus" mean. I would consider a limit to be using calculus, even if the original author didn't refer to these as limits, or justify them rigorously, etc. In the end, this is ultimately a completely subjective question. Nevertheless, I would be excited to see a proof for the area of a circle which did not require overtly taking any limits.

I disagree. We can talk about lots of irrational numbers without referring to Cauchy sequences or limits. For example, we can prove that $\sqrt{2}$ is irrational but this doesn't require calculus.

Even the definition of $\pi$ as the ratio of the circumference to diameter doesn't require calculus since you can prove this ratio is constant for any circle using just geometry.

I think the part of this question that is intimately tied to calculus is not the irrationality of $\pi$, but rather the fact that the notion of area itself is intimately tied to calculus. For example, you can choose an ellipse with rational axes and rational perimeter, but I still don't know how to compute its area without calculus.

Micrm@ss

Sure, how would you define area without calculus? And length?

studiot

Although he doesn't quite do that one Mark Levi tackles something more ambitious in his delightful book

The Mathematical Mechanic

Using Physical Reasoning to solve (Mathematical) Problems

skipjack

I think romsek meant that a regular n-gon of "radius" $r$ has area $A_n = nr^2\!\sin^2\left(\dfrac \pi n\right) \cot\left(\dfrac \pi n\right)$.

v8archie

Me too, which is why I deleted my post before you posted your reply.

I think the transcendental nature of $\pi$ might fit the bill though.

Area does not require calculus: see a square, for example. We can define area perfectly well without calculus. It is only certain types of area that require calculus: specifically those with curved boundaries. And the only reason we require it then is that our (only) method involves taking the limit of polygons.