Microwave penetration and losses are derived for the anisotropic norma
l and superconducting states of single crystals in the shape of thin p
latelets oriented parallel and perpendicular to the oscillating electr
omagnetic field. For platelet crystals with the microwave field parall
el to the major flat faces, the large anisotropy in the -normal state
can result in dissipation dominated by microwave field penetration thr
ough the thin edges rather than across the main faces. The influence o
f the extreme anisotropy is also considered for the superconducting st
ate and can account for an anomalous peak in microwave loss below T(c)
sometimes observed in Bi-Sr-Ca-Cu-O crystals. When crystals are mount
ed with their flat faces perpendicular to the microwave field, the los
ses in both the normal and superconducting states are shown to be stro
ngly peaked towards the outer perimeter of the crystals. This makes cr
itical demands on the degree of perfection of such regions, if the mic
rowave measurements are not to be complicated by nonintrinsic effects
associated with local imperfections.