  Modern cosmology is closely linked to our understanding of radial null
geodesics as these model the propagation of light signals through an expanding
universe. Azimuthal geodesics, on the other hand, are perhaps best known for
their relevance within closed cosmological models. Such models typically have a
finite lifetime: the universe expands up to a maximum size after which it
recollapses during the so-called big crunch. An azimuthal geodesic starting at
the beginning of the universe will travel a finite angular distance during the
expansion and recollapse. It is well-known that this angle is $2\pi$ for a
matter-dominated universe and $\pi$ for a radiation-dominated solution. Here we
derive the simple formula $$\Delta \varphi = \frac{2\pi}{1+3w}$$ for an
arbitrary linear equation of state parameter $w$. To the best of our knowledge
this result has not been reported elsewhere and fills a small gap in the
literature.
