Concentric circles of equal areas

Question

Here is the problem:

I have an annulus with radii r_i (inner radius) & R_o (outer radius). Knowing the two radii, we can compute an area of $A=\pi(R_o^2-r_i^2)$.

Now, divide A by n (A_t=A/n). Given A_t, find the (n-1) intermediate radii such that you end up with n concentric circles, each with an area of A_t.

This isn't that hard to do, the first intermediate radius outward from r_i is $R_{i1}=\sqrt{(A_t/\pi)+r_i^2}$, and $R_{i2}=\sqrt{(A_t/\pi)+R_{i1}^2}$, etc. with each previous radius value feeding into the next until all (n-1) intermediate radii are found.

My question, is there a proportional relationship between the intermediate radii? Say I am setting up a function in a program that needs to find the intermediate radii values, and all the user will give is r_i, R_o, and n. I could write a program that finds A, A_t, and R_i1 … R_{i n-1}, in order, using the previous inner radius to find the next intermediate outer radius.

But I think there is a more straightforward, proportional relationship, one where if I have what the user has given me, then the intermediate radii can be defined by R_{i x} = f(A, n), where it's not necessary to find the pervious intermediate radius first. Or something similar (maybe it's R_{i x} = f(A_t, n), or R_{i x} = f(A_t, r_i, n), etc.). Where the x in the subscript is whatever intermediate radius you are trying to find.

Example:

r_i = 0.2; R_o = 0.6; n = 5

A = 1.0053; A_t = 0.20106

R_{i 1-4} = 0.32249, 0.409878, 0.481664, 0.544059

Obviously the distance between the radii is getting smaller as the values head outward (increasing radius). But are the radii values decreasing by a proportional value that I can compute?

I think there is, and I am led to believe this from this comment by Professor Honeycomb on Reddit, where he links to this calculator page, which shows a relationship ($r_2=r_1\sqrt{2}$) between the area of an inner full circle and the annulus that surrounds it. But in my problem, I don't actually care about the area of the inner circle, it's the total area of the initial annulus I need to subdivide.

I also found the graphic in the lower right hand corner of this page, showing how to construct annulus of equal areas, which again suggest to me some kind of negative growth factor for the value of the radii, but I just can not see my way to an answer.

Thank you.

Answer 1

So we have an annulus of inner radius $r_i$, outer radius $R_o$, which we split into $n$ equal-area annuli, and we want the outer radius $r(p)$ of annulus $p$, where $p \in \{1,...,n\}$.

We want the area between $r_i$ and $r(p)$ to be $\frac p n$ of the area between $r_i$ and $R_o$. That is:
$\pi(r(p)^2-r_i^2)=\frac p n \pi(R_o^2-r_i^2)$.
Which gives $r(p)=\sqrt{r_i^2+\frac p n (R_o^2-r_i^2)} = r_i \sqrt{1+\frac p n ((\frac {R_o}{r_i})^2 - 1)}$.