  We study the geometry of hyperbolic knots that admit alternating projections
on embedded surfaces in closed 3-manifolds. We show that, under mild
hypothesis, their cusp area admits two sided bounds in terms of the twist
number of the alternating projection and the genus of the projection surface.
As a result, we derive diagrammatic estimates of slope lengths and give
applications to Dehn surgery. These generalize results of Lackenby and Purcell
about alternating knots in the 3-sphere.
  Using a result of Kalfagianni and Purcell, we point out that alternating
knots on surfaces of higher genus, can have arbitrarily small cusp density, in
contrast to alternating knots on spheres whose cusp densities are bounded away
from zero due to Lackenby and Purcell.
