An octagon is inscribed in a square so that the vertices of the octagon trisect the sides of the square. The perimeter of the square is 108 centimeters What is the perimeter of the octagon?
+592 $\dfrac{104}{3\cdot4}=9$ which is the length of one of the octogon's side lengths, and the diagonals are 45-45-90 triangles, so the other side of the octogon is $3\sqrt2$. so, $(3+3\sqrt2)\cdot4=\boxed{12+12\sqrt8}$
