Since the flag shown is rectangular, then its total area is its height multiplied by its width, or $h \times 2h=2h^{2}$. Since the flag is divided into seven stripes of equal height and each stripe has equal width, then the area of each stripe is the same. Since the four shaded strips have total area $1400 \mathrm{~cm}^{2}$, then the area of each strip is $1400 \div 4=350 \mathrm{~cm}^{2}$. Since the flag consists of 7 strips, then the total area of the flag is $350 \mathrm{~cm}^{2} \times 7=2450 \mathrm{~cm}^{2}$. Since the flag is $h$ by $2h$, then $2h^{2}=2450 \mathrm{~cm}^{2}$ or $h^{2}=1225 \mathrm{~cm}^{2}$. Therefore, $h=\sqrt{1225 \mathrm{~cm}^{2}}=35 \mathrm{~cm}$ (since $h>0$). The height of the flag is 35 cm.