Why the Area of a Circle is Equal to πr^2?

I have been told that the area of a circle is equal to \pi r^2 since I was in primary school by my math teacher, and from then on I use this formula to solve many problems about circles. And these days, when I used this formula again, I suddenly realized that I didn’t know why the formula will look like this in fact.

So I began to think how to get this formula. In my opinion, it’s related to the limit, so I proved the formula by calculus later.

Consider that what does \pi stand for. It’s the ratio of a circle’s circumference to its diameter. Let C be a circle’s circumference and d be its diameter. We have

\displaystyle \pi=\frac{C}{d} \ \ \ (\ast)

And we can divide the circle into many circles whose diameter is not bigger than C. And when the number of small circles approaches infinity, we can calculate the area of the big circle by integral. Let r be the radius of the circle, we have C=\pi d=2\pi r according to (\ast).

Therefore

\displaystyle S=\int_{0}^{r}2\pi x\ dx=\pi r^2

What’s more, the result is also tenable in a ball.

We can get the surface area of a ball S=4\pi r^2 by calculus.

And then do it as above, we can figure out the volume of the ball

\displaystyle V=\int_{0}^{r}4\pi x^2\ dx=\frac{4}{3}\pi r^3

Although the result is clear, it’s still valuable for us to think why it can be this. And sometimes when I meet something obvious but hard to understand (or never think how to understand it), I will have a try and think it over.

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