Okay, let's see. I need to find the length of the diagonal of a square when its perimeter is equal to that of an equilateral triangle, and the area of the triangle is given as 16√3 square centimeters. Hmm, let me break this down step by step.

First, I remember that the perimeter of a shape is the sum of all its sides. For an equilateral triangle, all three sides are equal, so if I can find the length of one side, I can easily find the perimeter. Since the area is given, maybe I can use that to find the side length.

The formula for the area of an equilateral triangle is (√3/4) * side². Let me write that down:

Area = (√3 / 4) * s²

We know the area is 16√3, so plugging that in:

16√3 = (√3 / 4) * s²

Hmm, let's solve for s. First, I can multiply both sides by 4 to get rid of the denominator:

4 * 16√3 = √3 * s²

Which simplifies to:

64√3 = √3 * s²

Now, divide both sides by √3 to isolate s²:

64√3 / √3 = s²

The √3 terms cancel out, so:

64 = s²

Taking the square root of both sides gives:

s = 8 cm

So, each side of the equilateral triangle is 8 centimeters. Since it's an equilateral triangle, the perimeter is 3 times the side length:

Perimeter of triangle = 3 * 8 = 24 cm

The problem states that the square has the same perimeter. So, the perimeter of the square is also 24 cm. The perimeter of a square is 4 times the side length, so:

Perimeter of square = 4 * side = 24 cm

Therefore, the side length of the square is:

side = 24 / 4 = 6 cm

Now, the question asks for the length of the diagonal of the square. I recall that in a square, the diagonal forms a right triangle with two sides of the square. So, using the Pythagorean theorem, the diagonal (d) can be found as:

d = √(side² + side²) = √(2 * side²) = side * √2

Substituting the side length we found:

d = 6 * √2 cm

Let me just double-check my steps to make sure I didn't make any mistakes. Starting from the area of the triangle, solved for the side, got 8 cm, then perimeter 24 cm. Square side is 6 cm, diagonal is 6√2 cm. That seems right. The area formula for the triangle was correct, and the perimeter calculations follow. Yeah, I think that's all. So the diagonal should be 6√2 centimeters.

**Final Answer**
The length of the diagonal of the square is \boxed{6\sqrt{2}} centimeters.
Therefore, the diagonal of the square is $\boxed{6\sqrt{2}}$ centimeters.
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