  In this paper we consider random planar maps weighted by the self-dual
Fortuin--Kasteleyn model with parameter $q \in (0,4)$. Using a bijection due to
Sheffield and a connection to planar Brownian motion in a cone we obtain
rigorously the value of the critical exponent associated with the length of
cluster interfaces, which is shown to be $$
  \frac{4}{\pi} \arccos \left( \frac{\sqrt{2 - \sqrt{q}}}{2}
\right)=\frac{\kappa'}{8}. $$ where $\kappa' $ is the SLE parameter associated
with this model. We also derive the exponent corresponding to the area enclosed
by a loop which is shown to be 1 for all values of $q \in (0,4)$. Applying the
KPZ formula we find that this value is consistent with the dimension of SLE
curves and SLE duality.
