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.
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.
Best Answer
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 $r$