Okay, let me try to figure out this problem. So, we have a regular octagon and a regular hexagon. The hexagon has a side length of 16 cm, and both shapes have the same perimeter. The question is asking for the length of each side of the octagon. Hmm, okay. Let's start by recalling some basic facts about regular polygons.

First, a regular polygon has all sides equal and all angles equal. The perimeter of a regular polygon is just the number of sides multiplied by the length of one side. So, for the hexagon, since it's regular, its perimeter would be 6 times the side length. The hexagon's side length is given as 16 cm, so the perimeter of the hexagon is 6 * 16 cm. Let me calculate that: 6 * 16 is 96, right? So the perimeter is 96 cm.

Now, the octagon has the same perimeter as the hexagon. So, the perimeter of the octagon is also 96 cm. A regular octagon has 8 sides, so if I let 's' be the length of each side of the octagon, then the perimeter would be 8 * s. Therefore, 8s = 96 cm. To find 's', I just need to divide both sides by 8. So, s = 96 / 8. Let me compute that: 96 divided by 8 is 12. So, each side of the octagon should be 12 cm long.

Wait, that seems straightforward. Let me double-check my steps to make sure I didn't make any mistakes. First, perimeter of the hexagon: 6 sides * 16 cm = 96 cm. Then, the octagon's perimeter is equal to that, so 8s = 96. Solving for s gives 12 cm. Hmm, that seems right. But let me verify if there's any trick or something I missed here.

The problem mentions an Asymptote figure, but since I can't see the image, the Asymptote code is provided. Let me parse that code to see if it gives any clues. The Asymptote code defines points A and B as directions 120 and 60 degrees, respectively. Then it draws lines connecting dir(360), B, A, dir(180), dir(240), dir(300), and back to the start. The label is placed midway between A and B with the length 16 cm. Since this is a regular hexagon, each side should indeed be 16 cm. Wait, but in the Asymptote code, the hexagon is drawn with specific angles. Let me check if that's a regular hexagon or not. But since it's mentioned as a regular hexagon with side length 16 cm, the perimeter calculation should hold.

Therefore, I think my initial calculation is correct. The side length of the octagon is 12 cm. I don't see any errors in the reasoning. The key steps are calculating the perimeter of the hexagon and then dividing by the number of sides of the octagon to get the side length. Since both perimeters are equal, this should give the correct answer. So, I'm confident that the answer is 12 cm.
Therefore, each side of the octagon is $\boxed{12}$ centimeters long.
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