<Question>
A square and an equilateral triangle have	equal	perimeters.	The area of the triangle is $16\sqrt{3}$ square centimeters. How long, in centimeters, is a diagonal of the square? Express your answer in simplest radical form.

[asy]
defaultpen(linewidth(1));
draw((0,0)--(1,0)--(1,1)--(0,1)--cycle);
pair a = (1.25,0)+1.25*dir(60);
pair b = a+1.25*dir(-60);
draw((1.25,0)--a--b--cycle);

[/asy]
</Question>

<think>
<Plan_1>
Okay, so I need to find the length of the diagonal of a square, given that the square and an equilateral triangle have equal perimeters. The area of the triangle is 16√3 square centimeters. Hmm, let me break this down step by step.

<prompt_1.1> Parse the problem to identify the key elements and what needs to be solved. </prompt_1.1>
</Plan_1>

<Execution_1>
<execution_1.1>
First, let me recall some formulas. The perimeter of a square is 4 times the side length, and the perimeter of an equilateral triangle is 3 times its side length. Since their perimeters are equal, if I can find the side length of the triangle, I can find the perimeter, then divide by 4 to get the side length of the square. Once I have the side length of the square, I can find the diagonal using the formula for the diagonal of a square, which is side length times √2. That should give me the answer.

But first, I need to find the side length of the equilateral triangle. The problem gives the area of the triangle as 16√3 cm². I remember that the area of an equilateral triangle can be calculated using the formula (√3/4) * side². Let me confirm that formula.
</execution_1.1>
</Execution_1>

<Plan_2>
Based on execution_1.1:
Yes, for an equilateral triangle, all sides are equal, and the height can be found using Pythagoras, which leads to the area formula. So if the area is 16√3, I can set up the equation:

<prompt_2.1> Use the area formula for an equilateral triangle to find its side length. </prompt_2.1>
-----

But let me make sure that the perimeters are indeed equal. Let me compute the perimeter of the triangle, which is 3*8=24 cm.

<prompt_2.2> Verify that the perimeters of the square and triangle are equal. </prompt_2.2>
-----

But let me just check once more if I interpreted the problem correctly. The problem says the area of the triangle is 16√3.

<prompt_2.3> Confirm the interpretation of the problem and solution. </prompt_2.3>
</Plan_2>

<Execution_2>
<execution_2.1>
(√3/4) * (side length of triangle)² = 16√3

Let me solve for the side length. Let's denote the side length of the triangle as 'a'. Then:

(√3/4) * a² = 16√3

Hmm, I can multiply both sides by 4/√3 to isolate a². Let's try that:

a² = 16√3 * (4/√3)

Wait, that would be:

a² = (16√3) * (4 / √3) = 16 * 4 * (√3 / √3) = 64 * 1 = 64

So a² = 64, which means a = 8 cm. So the side length of the triangle is 8 cm.
</execution_2.1>

<execution_2.2>
Then the perimeter of the square should also be 24 cm. Therefore, each side of the square is 24/4 = 6 cm. Then the diagonal of the square would be 6√2 cm.
</execution_2.2>

<execution_2.3>
Yes, that's given. So using the area formula for an equilateral triangle gives the side length 8, perimeter 24, square side 6, diagonal 6√2. All steps check out.
</execution_2.3>
</Execution_2>

<Plan_3>
Based on execution_2.1:
Wait, let me check that again. If I have (√3/4) * a² = 16√3, then multiplying both sides by 4/√3 gives a² = (16√3)*(4/√3).
Wait, hold on, that would be 16√3 * 4 / √3. The √3 in the numerator and denominator cancels out, so 16*4=64.

<prompt_3.1> Verify the calculation of the side length of the triangle. </prompt_3.1>

Then the perimeter of the triangle is 3*8=24 cm. Therefore, the perimeter of the square is also 24 cm. So each side of the square is 24 divided by 4, which is 6 cm.
Then, the diagonal of a square is side length times √2, so 6√2 cm.

Wait, that seems straightforward, but let me make sure I didn't skip any steps or make any miscalculations.
-----

So the area formula for an equilateral triangle is indeed (√3/4)*a². Plugging in a=8, we get (√3/4)*64 = 16√3, which matches the given area.
</Plan_3>

<Execution_3>
<execution_3.1>
So a² = 64, so a = 8 cm. That seems right. So the triangle has sides of 8 cm each, area 16√3 cm².
</execution_3.1>
</Execution_3>

<Plan_4>
Based on execution_3.1:
Therefore, the perimeter of the triangle is 3*8=24 cm. Since the square has the same perimeter, each side of the square is 24/4=6 cm.

<prompt_4.1> Calculate the diagonal of the square using its side length. </prompt_4.1>

But let me double-check all steps to be sure. Let's start from the area of the triangle again. The formula for the area of an equilateral triangle is (√3/4)*a². If the area is 16√3, then:

<prompt_4.2> Re-evaluate all steps to confirm the solution. </prompt_4.2>
</Plan_4>

<Execution_4>
<execution_4.1>
Then, the diagonal of a square is given by the formula s√2, where s is the side length. So 6*√2 cm is the diagonal. Therefore, the answer should be 6√2 centimeters.

Wait, but let me confirm once more. Let's retrace all steps:

1. Area of equilateral triangle = 16√3 = (√3/4)*a².
2. Solving for a: a² = (16√3)*(4/√3) = 64 => a=8.
3. Perimeter of triangle = 3*8=24.
4. Perimeter of square = 24 => side of square = 24/4=6.
5. Diagonal of square = 6√2.
</execution_4.1>

<execution_4.2>
(√3/4)*a² = 16√3

Multiply both sides by 4:

√3 * a² = 64√3

Divide both sides by √3:

a² = 64

So a = 8. Correct. Then perimeter is 3*8=24. Square's perimeter is also 24, so each side is 24/4=6. Then diagonal of square is 6√2.
</execution_4.2>
</Execution_4>

<Final_answer>
Therefore, the length of the diagonal of the square is $\boxed{6\sqrt{2}}$ centimeters.
</Final_answer>
</think>