  In this paper we study the area requirements of planar greedy drawings of
triconnected planar graphs. Cao, Strelzoff, and Sun exhibited a family $\cal H$
of subdivisions of triconnected plane graphs and claimed that every planar
greedy drawing of the graphs in $\mathcal H$ respecting the prescribed plane
embedding requires exponential area. However, we show that every $n$-vertex
graph in $\cal H$ actually has a planar greedy drawing respecting the
prescribed plane embedding on an $O(n)\times O(n)$ grid. This reopens the
question whether triconnected planar graphs admit planar greedy drawings on a
polynomial-size grid. Further, we provide evidence for a positive answer to the
above question by proving that every $n$-vertex Halin graph admits a planar
greedy drawing on an $O(n)\times O(n)$ grid. Both such results are obtained by
actually constructing drawings that are convex and angle-monotone. Finally, we
consider $\alpha$-Schnyder drawings, which are angle-monotone and hence greedy
if $\alpha\leq 30^\circ$, and show that there exist planar triangulations for
which every $\alpha$-Schnyder drawing with a fixed $\alpha<60^\circ$ requires
exponential area for any resolution rule.
