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Alternatively, you can prove that the area of any circle is $\pi r^2$ entirely by computing the definite integral $\int_{-r}^r 2\sqrt{r^2 - x^2} dx$ without even making the assumption that the area of any circle varies as the square of its radius.
I understand the reasoning behind $\pi r^2$ for a circle area however I'd like to know what is wrong with the reasoning below: The area of a square is like a line, the height (one dimension, length) placed several times next to each other up to the square all the way until the square length thus we have height x length for the area.
Hint: Try subtracting $2\pi r^2 + 2\pi rh$ from both sides of the original and completing the square. I can expand on this if you need me to. I can expand on this if you need me to. Share
$\begingroup$ This understates the subtlety of Archimedes' proof. It not only involves using limits to calculate areas (even Euclid did that!), but also requires a convexity argument which — if fully justified for curved figures — will end up using some notion of tangent line.
I was looking for proofs using Calculus for the area of a circle and come across this one $$\\int 2 \\pi r \\, dr = 2\\pi \\frac {r^2}{2} = \\pi r^2$$ and it struck me as being particularly easy. The only
Now as one increases the number of sides of the polygon indefinitely, its area and circumference both tend to those of the circle. So the area of the circle should also be $\frac r2$ times its circumference, which circumference is $2\pi r$ by definition, and this gives an area of$~\pi r^2$.
One geometric explanation is that $4\pi r^2$ is the derivative of $\frac{4}{3}\pi r^3$, the volume of the ...
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$\begingroup$ An elementary proof does not exist. I seem to recall a piece in one of MAA's publications (from the mid 1990s---this may be it) which points out this issue, and I have colleague who has written a pretty detailed paper which tries to patch things together using the techniques available to Archimedes.
A number of different ways of showing this exist. First notice that the area has to be $(\text{constant}\cdot r^2)$ because the area of a region of any shape in a plane must be proportional to the square of the distances.