680. The sides of three regular octagons are respectively equal to 3 l, $4 \boldsymbol{\mu}, 12 \boldsymbol{}$. What is the length of the side of a regular octagon whose area is equal to the sum of the areas of the three given octagons?
680. Let the ratio of linear measurements of three similar figures $F_{1}, F_{2}, F_{3}$ (in this case, regular octagons) be $k: l: m$ (in this case $3: 4: 12$). Denote the areas of the figures as $S_{1}, S_{2}, S_{3}$, and let $F_{4}$ be a figure similar to $F_{1}, F_{2}, F_{3}$ with area $S_{4}=S_{1}+S_{2}+S_{3}$.

Since $S_{1}: S_{2}: S_{3}: S_{1}=k^{2}: l^{2}: m^{2}:\left(k^{2}+l^{2}+m^{2}\right)$, the linear measurements of the figures $F_{1}, F_{2}, F_{3}, F_{4}$ are in the ratio $k: l: m: V \sqrt{k^{2}+l^{2}+m^{2}}$ (in our case $3: 4: 12: 13$). Therefore, the side of the required octagon is $13 m$.