解:∵$∠B = 90°,$$AB = 3,$$BC = 4$
∴$AC^2=AB^2+BC^2=25,$$AC=5$
$ $在$\triangle ACD$中,$AC^2=25,$$CD = 12,$$AD = 13$
∵$AC^2+CD^2=25+12^2= 169,$$AD^2=13^2=169$
∴$AC^2+CD^2=AD^2$
∴$\triangle ACD$是直角三角形,且$∠ACD = 90°$
∴$S_{四边形ABCD}= S_{\triangle ABC}+S_{\triangle ACD}$
$ =\frac 12×AB×BC+\frac 12×AC×CD$
$ =\frac 12×3×4+\frac 12×5×12=36$