TITLE: How is the area of a circle calculated using basic mathematics?
QUESTION [14 upvotes]: Area of a circle is addition of circumference of layers of a onion. If n is radius of a onion then area is
$$
A = 2 \pi \cdot 1 + 2 \pi \cdot 2 + 2\pi \cdot 3 + \ldots + 2  \pi \cdot n
$$
which 
$$
= 2 \pi \cdot \frac{n(n+1)}{2}
= \pi (n^2 + n)
$$
But the answer is wrong. Please tell me where am I wrong?

REPLY [2 votes]: You are not so far: the correct surface is $\pi n^2$. The ratio of your formula to the expected area is
$$\frac{n^2+n}{n^2}=1+\frac1n.$$
As you must take $n$ as large as possible, the ratio is $1$.

Note that your computation is approximate as your layers have a thickness of $1$ and their area is that of a ring, $\pi((n^2-(n-1)^2)=\pi(2n-1)$ and not $2\pi$, but as you increase $n$, the difference becomes negligible.