Conformal modulus of an annulus ($M=(2\pi {)}^{-1}\log (R/r)$)

The height divided by circumference in a conformal Euclidean cylinder model of an annulus. For $r<|z|<R$, it is $(2\pi {)}^{-1}\log (R/r)$. The extremal length of winding-one core curves is $1/M$, whereas that of curves joining the two boundary components is $M$.

For an inner circle of real center $c$ and radius $d$ lying strictly inside the unit circle, choose the real root $|a|<1$ of $c{a}^2-(1+c^2-d^2)a+c=0$. When $c=0$, take $a=0$. The automorphism of the unit disk $T_a$ sends the inner circle to a concentric circle. Indeed it sends the two real endpoints $c-d,c+d$ to opposite real numbers; preservation of circles and reflection symmetry then give an image centered at zero. Its radius is $r=T_a(c+d)\in (0,1)$.

The conformal invariance of harmonic measure and harmonic measure of an annulus give outer-boundary measure $\log (|T_a(z)|/r)/\log (1/r)$. For $c=d=2/5$, the quadratic gives $a=1/2$ and $r=1/2$, an explicit eccentric-annulus example.

The quotient of a horizontal strip by a horizontal translation. The model $(\mathbb{R}/\mathbb{Z})\times (0,M)$ has conformal modulus of an annulus $M$. Averaging horizontal loop lengths and applying Cauchy-Schwarz inequality proves that the core-loop extremal length equals $1/M$.