I know that the equation for a torus is given by $$\left(R - \sqrt{x^2 + y^2}\right)^2 + z^2 = r^2$$
where $R$ is the larger radius and the $r$ is the smaller radius. Now I was reading about the solid torus, which can be constructed by forming a cartesian product of a disc in $D^2 \in \mathbb{R}^2$ and a circle $S^1$. Suppose for example my disc has radius $r$ and the circle has radius $R$, then what does the equation representing a solid torus look like?
On
If you carefully read the article that you have cited, you will notice that it says the solid torus is homeomorphic to the Cartesian product $S^1\times D^2.$ To say two things are homeomorphic is quite different from saying one is a construction of the other.
Notice that the same page tells you that the ordinary torus is homeomorphic to $S^1\times S^1.$ If you can explain how this statement leads directly to the equation $\left(R - \sqrt{x^2 + y^2}\right)^2 + z^2 = r^2$ then I think you should write your own answer to your question about the solid torus. I can only get to that equation by an indirect route shown below.
Since the solid torus is homeomorphic to $S^1\times D^2,$ however, that gives us some clues about how we might parameterize a solid torus. We can use the Cartesian product of a parameterization of a disk with a parameterization of a circle.
If we take a disk of radius $r$ as a region in a plane with Cartesian coordinates $(u,v)$ that satisfy $u^2 + v^2 \leq r^2,$ then the coordinates $(u,v)$ are a reasonable parameterization of the disk.
Cartesian coordinates are not a good parameterization of a circle. A better parameterization is the angle $\theta$ from some reference point on the circle. So if we have a circle of radius $R$ in a plane with Cartesian coordinates $(s,t),$ you can find any point on the circle by setting \begin{align} s &= R \cos\theta, \tag1\\ t &= R \sin\theta. \tag2\\ \end{align} You can easily confirm that then $s^2 + t^2 = R^2.$ But when it comes to naming points on the circle, it is better to forget about the coordinates $(s,t)$ and just use $\theta$ as a coordinate (that is, just the angle component of polar coordinates). This is because the circle is a one-dimensional object and should have just one coordinate.
We end up then with coordinates $(u,v,\theta)$ for the solid torus, with the "equation" (actually inequation) $$ u^2 + v^2 \leq r^2.$$
If we write the coordinates in the sequence $(\theta,u,v)$ instead then we have parameterized the solid torus literally according to its homeomorphism with $S^1\times D^2.$ That is, we have a parameter of $S^1$ concatenated with parameters of $D^2.$
If the $v$ axis of the disk at each $\theta$ is aligned with the axis of rotation of the torus and the positive $u$ direction is oriented away from the axis, then these are (almost) the cylindrical coordinates for a torus. To get the usual cylindrical coordinates we merely need to set $\rho = u + R$ and then the inequation of the solid torus in cylindrical coordinates $(\rho,v,\theta)$ is $$ (\rho - R)^2 + v^2 \leq r^2, $$ Note the similarity to the Cartesian equation for the torus, setting $\rho = \sqrt{x^2 + y^2}$ and $v = z.$ Indeed, an equation for the torus (the boundary of the solid torus) in cylindrical coordinates is $$ (\rho - R)^2 + v^2 = r^2 $$ or equivalently $(R - \rho)^2 + v^2 = r^2.$ The substitution $\rho = \sqrt{x^2 + y^2}$ and $v = z$ then turns this equation into the known equation $\left(R - \sqrt{x^2 + y^2}\right)^2 + z^2 = r^2.$
If you insist on using Cartesian coordinates for both $S^1$ and $D^2$ in the Cartesian product $S^1\times D^2$ and applying those same coordinates to the solid torus, keep in mind that for some objects, such as an arbitrary straight line in three-dimensional space with Cartesian coordinates, there is no single literal equation that describes the object. The "equation" of the object may actually consist of several equations or a combination of equations and inequations; for example, the pair of equations \begin{align} x + y &= 2, \\ x - z &= 1 \\ \end{align} describe a particular line in $(x,y,z)$ coordinates.
Using $s^2 + t^2 = R^2$ as an equation for a circle $S^1$ and $ u^2 + v^2 \leq r^2$ as an inequation for a disk $D^2,$ then an "equation" for the solid torus can simply be the combination of that equation and inequation, \begin{align} s^2 + t^2 &= R^2, \\ u^2 + v^2 &\leq r^2. \\ \end{align}
As a description of a torus this leaves a lot to be desired. For one thing, it gives us four coordinates, $(s,t,u,v)$ for an object that exists in three-dimensional space. We could take the view that the $s$ and $t$ specify a coordinate $\theta$ as the simultaneous solution of Equations $(1)$ and $(2)$.
Some parametrization is helpful.
$$\rho= R+r \cos\phi\quad z= r \sin \phi \;$$
$$(\rho-R)^2+z^2=(R-\sqrt{x^2+y^2})^2 +z^2=r^2 $$
We can employ a variable along the normal to the circle section $ 0< rt<r $. Then the solid is the sum of many layers of expanding $r$ with $ 0<t<1 :$
$$\left(R - \sqrt{x^2 + y^2}\right)^2 + z^2 = r^2 t^2$$
Simply put, this exterior surface is associated with the torus. If the inside of the toroidal surface is filled with a fluid then the volume is associated with the solid torus.
Sections on Torus show some outer profiles of the layered solid when sectioned along a radius.
Parts of sections of such tori can be seen through oval annular rings in lumber/plywood/studs around knots and single cut bent branches.