The authors introduce a new definition of a cluster's accessible external perimeter, which is different from the usual definition of the 'hull'. This external perimeter can be measured by probing (adsorbent) particles coming from the cluster's exterior. The size of these particles determines which perimeter sites are accessible from the outside and hence one can define different 'external' or 'accessible' perimeters according to this size. Using a Monte Carlo simulation they measured the fractal dimensions of these accessible perimeters, De, for percolation clusters (at the threshold) on the square and the triangular lattices. The fractal dimension of the accessible perimeter is found to depend on the size of the probing particles. For several sizes which are larger than some (lattice-dependent) threshold they find D e approximately=4/3, which is significantly smaller than the 'hull dimension', Dh=7/4 (found with smaller probing particles).