# Why is the slope of stacked concentric circles 2*pi*r? (About a calculus video by 3Blue1Brown.)

View the video here.

In the video, Grant, its creator, breaks down the area formula of a circle.

Explaining the video as best I can:

He divides the circle into numerous smaller, concentric circles, each with a thickness dr and length 2 x pi x R. (He uses the r notation again, but I'll go with R so as not to confuse the two.)

He then unravels the circles into lines and approximates them with rectangles, the area of which we can easily compute with dr x (2 x pi x R). After that, Grant places each line on an x-y graph, where the x axis represents the thickness (dr) and the y the length (2 x pi x R).

He runs a line from origin over the triangle-ish shape formed by these figures (hard to explain, please see the video) and says its slope is 2 x pi x r. Here, I get stuck.

According to my own calculations — and I may be wrong here — the formula for R is r / (N - n), where N is the total number of concentric circles and n is the distance of a specific circle from the center.

So, to find the slope, I plugged two pairs of values calculated with that formula and simply dr into the standard slope formula — (y_1 - y_2) / (x_1 - x_2) — and got (pi x r) / 3 x dr. If I plug in the value of dr in this example, which is 3/4 (I have 4 concentric rings), I get (4 x pi x r)/9, which still isn't right.

Here's how.

I know I went wrong somewhere. But I don't know whether there's a computational error I made along the way that I cannot find or if I'm misunderstanding something in the video.

• Grant drew the graph of $y=2\pi r$ and said the slope was $2\pi$. That is at 5:26 to 5:29 in his video. The x-axis is R (or r. I'm not sure how you see them as different.) Your mistake might be in assuming the x-axis is $dr$, when it is really $r$ – Narlin Sep 21 '18 at 19:42
• Yes, that must be it! I both kept looking at the x axis in term of dr and assumed the slope was 2 x pi x r, because it was written so above it. What a stupid problem. Will you post that as an answer? I'll mark it as correct. Tomorrow. (Going to sleep, now. Timezones.) – Demetre Saghliani Sep 21 '18 at 20:03

Grant drew the graph of y=2πr and said the slope was 2π. That is at 5:26 to 5:29 in his video. The x-axis is R (or r. I'm not sure how you see them as different.) Your mistake might be in assuming the x-axis is $$dr$$, when it is really $$r$$