This should be a simple, known result, but I can't seem to find it.
Given a lattice $\Gamma = m\mathbb{Z} \times n\mathbb{Z}$, $\mathbb{R}^2/\Gamma$ is topologically a torus. For suitable $m$ and $n$ (say $m$ big enough and $n$ small enough), this torus can be embedded in $\mathbb{R}^3$ by the parametrization
$x(\theta,\phi) = ((R+r\cos\theta)\cos\phi,(R+r\cos\theta)\sin\phi,r\sin\theta).$
Without loss of generality, we can take $n = 1$ and $m > 1.$ Given $m$ and $1$, what are the values of $R$ and $r$?
If we consider the topological construction, we can say that we identify the long edges so that the small circle of the obtained cylinder has radius $n/2\pi$. However, identifying the remaining sides will create stretching so that we can no longer say the radius is $m/2\pi$.
Alternatively, we have a torus in $S^3$ given by $x(\theta,\phi) = (\sin\rho\cos\theta,\sin\rho\sin\theta,\cos\rho\cos\phi,\cos\rho\sin\phi),$ where $\rho$ is a parameter that allows us to determine a torus with any ratio of radii. Is it true that $m/n = \sin\rho$ (or something like that)? Seems so; how can I show it?
I have a conjecture that $R = \sqrt{m^2 + n^2}$ but don't know how to show it.
The point is to identify any torus in $\mathbb{H}/SL_2(\mathbb{Z})$ with a parametrization so that I may find the area and volume and the energy of a certain functional (Willmore). Does anyone know perhaps simpler ways of determining area and volume given a point in the typical fundamental domain of the modular surface?