Defining the equatorward boundary of the cusp region in the ionosphere as the projection of the merging line from the magnetopause, we use a quantitative, geometrically realistic model to show how the local time span of the cusp increases with increasing merging rate for southward interplanetary magnetic field. Since the merging rate is fixed by the magnitude of the magnetic field component normal to the model magnetopause and then normalized to the cross-polar-cap potential, the result gives the variation of cusp local time span as a function of potential. From 0 kV up to 60 kV, the cusp expands from a point at noon to 2 hours on either side. Nearly tripling the voltage to 170 kV adds one more hour to each side, yielding a span from 0900 to 1500 LT. As an example of broad local time span during magnetically active periods, we present spacecraft observations of cusp particles during the great storm of March 1989 that cover more than 8 hours of local time. An important aspect of the result is the demonstration that a merging line of fixed length on the magnetopause, as assumed in the model, maps to a projected length in the ionosphere that increases as the funnel-shaped cusp opens. This behavior contrasts with earlier models that have cleft rather than cusp geometry, where the projected merging line length is proportional only to its length on the magnetopause. The model results are used to construct the footprint of a flux transfer event caused by time variations of the merging rate, uniform along the length of the merging line. The cusp geometry distorts the field lines mapped from the magnetopause to yield footprints with dawn and dusk protrusions into the region of closed magnetic flux. ¿ American Geophysical Union 1991