A cubical cake with edge length $6$ inches is iced on the top and the four sides. Label the corners of its top face $P,Q,R,S$ in order, and let $M$ be the midpoint of $\overline{PS}$. The cake is cut straight down along $\overline{MQ}$ and $\overline{MR}$, giving three pieces. The piece whose top is triangle $PMQ$ contains $c$ cubic inches of cake and $s$ square inches of icing. What is $c+s$?

1. Put $P=(0,0)$, $Q=(6,0)$, $R=(6,6)$, $S=(0,6)$, so $M=(0,3)$. The top of the piece is triangle $PMQ$ with base $PQ=6$ and height $PM=3$, so its area is $\frac12\cdot6\cdot3=9$.
2. The cake is $6$ inches tall, so the piece has volume $c=9\cdot6=54$.
3. Its icing is the top, $9$, together with the two outer side faces it touches: along $PM$ (length $3$) and along $PQ$ (length $6$), each $6$ inches tall, giving $3\cdot6+6\cdot6=54$. So $s=9+54=63$.
4. $c+s=54+63=\boxed{117}$.