In this paper we consider random planar maps weighted by the self-dual Fortuin--Kastelyn 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 critical exponents associated with the length of cluster interfaces, which is shown to be $$\frac{4}{\pi} arccos\left(\frac{\sqrt{2-\sqrt{q}}}{2}\right).$$ Similar results are obtained for the area. Applying the KPZ formula we find that this value is consistent with the dimension of SLE curves and SLE duality. Various isoperimetric relationships of independent interest are also derived.