I'm attempting to form interior structures of planets. If I know the volume percentage of each layer of the planet, how would I determine the outer radius of each layer?
Let's assume the planet is divided into three main layers: core, mantle, and crust. The core can be expressed as a sphere with some radius $R_c$. The mantle and crust can be expressed as spherical shells concentric to the core. The mantle has an outer radius $R_m$ and an inner radius $R_c$. Likewise, the crust has an outer radius $R_p$ and an inner radius $R_m$. $R_p$ is the radius of the planet overall.
I know the volume of the core can be expressed as $$V_c= \frac {4}{3}R_c^3\pi$$ and the volume of the mantle is $$V_m=\frac{4}{3}\pi(R_m^3-R_c^3).$$ Let's call the radius percentage for the core and mantle $r_c$ and $r_m$, respectively. $r_c=\frac{R_c}{R_p}$ and $r_m=\frac{R_m}{R_p}$. Likewise, the volume percentages for the core and mantle are $v_c$ and $v_m$, respectively. $v_c=\frac{V_c}{V_p}$ and $v_m=\frac{V_m}{V_p}$.
I'm pretty sure that $r_c=\sqrt[3]{v_c}$, though I'm not sure. How do I go about finding $r_m$? Would I need to know $r_c$ or $R_c$? Does the volume of the core matter for finding the mantle's outer radius?
You will find things clearer if instead of thinking about
you think about
The volume-fractions for these are just the cubes of the radius-fractions, because these things are spheres rather then spherical shells and the volume of a sphere is proportional to the cube of its radius.
So if you know the volume-fractions for core, mantle, crust then it's trivial to compute from these the volume-fractions for core, core+mantle, core+mantle+crust. And then from these you can get your radius-fractions by taking the cube root.
Concretely: suppose the volume-fractions for core, mantle, crust are $u,v,w$ (which of course have to add up to 1). Then the fractions I think you should pay attention to are $u,u+v,u+v+w=1$. Then the radius-fractions are $\root3\of{u},\root3\of{u+v},\root3\of{u+v+w}=1$. The middle one of those is your $r_m$. Whether it "depends on" the volume of the core is a matter of definitions; you can write it as $\root3\of{u+v}$ in which case it does, or as $\root3\of{1-w}$ in which case it doesn't.