$解:如图,在Rt△ABD中,由勾股定理得AB=\sqrt{3^{2}+4^{2}}=5$
$在Rt△ACF中,由勾股定理得AC=\sqrt{2^{2}+6^{2}}=2\sqrt{10}$
$在Rt△BCE中,由勾股定理得BC=\sqrt{2^{2}+5^{2}}=\sqrt{29}$
$∴△ABC的周长为5+2\sqrt{10}+\sqrt{29}$
$S_{△ABC}=S_{长方形CFDE}-S_{△ABD}-S_{△ACF}-S_{△BCE}$
$=5×6-\frac{1}{2}×3×4-\frac{1}{2}×2×6-\frac{1}{2}×5×2$
$=30-6-6-5$
$=13$