We give area and height estimates for cmc-graphs over a bounded planar $C^{2,{alpha}}$ domain ${Omega}{subset}mathbb{R}^3$. For a constant H satisfying $H^2{mid}{Omega}{mid}{leq}9{pi}/16$, we show that the height $h$ of H-graphs over ${Omega}$ with vanishing boundary satisfies ${mid}h{mid}$ < $( ilde{r}/2{pi})H{mid}{Omega}{mid}$, where $ ilde{r}$ is the middle zero of $(x-1)(H^2{mid}{Omega}{mid}(x+2)^2-9{pi}(x-1))$. We use this height estimate to prove the following existence result for cmc H-graphs: for a constant H satisfying $H^2{mid}{Omega}{mid}$ < $(sqrt{297}-13){pi}/8$, there exists an H-graph with vanishing boundary.