  We present inelastic neutron scattering measurements of the Cairo pentagon
lattice magnets Bi$_2$Fe$_4$O$_9$ and Bi$_4$Fe$_5$O$_{13}$F, supported by high
field magnetisation measurements of Bi$_2$Fe$_4$O$_9$. Using linear spin wave
theory and mean field analyses we determine the spin exchange interactions and
single-ion anisotropy in these materials. The Cairo lattice is geometrically
frustrated and consists of two inequivalent magnetic sites, both occupied by
Fe$^{3+}$ ions and connected by two competing nearest neighbour interactions.
We found that one of these interactions, coupling nearest neighbour spins on
the three-fold symmetric sites, is extremely strong and antiferromagnetic.
These strongly coupled dimers are then weakly coupled to a framework formed
from spins occupying the other inequivalent site. In addition we found that the
Fe$^{3+}$ $S=5/2$ spins have a non-negligible single-ion anisotropy, which
manifests as a spin anisotropy gap in the neutron spectrum and a spin-flop
transition in high field magnetisation measurements.
