We are given that the area of the square is 18 times the area of the triangle, and the dimensions of the triangle are provided as follows:

• Height of the triangle = $\frac{5}{8}$ cm
• Base of the triangle = $6 \frac{2}{5}$ cm = $6.4$ cm

Step 1: Calculate the area of the triangle

The area of a triangle is given by the formula:

$\text{Area of triangle} = \frac{1}{2} \times \text{base} \times \text{height}$

Substituting the given values:

$\text{Area of triangle} = \frac{1}{2} \times 6.4 \, \text{cm} \times \frac{5}{8} \, \text{cm}$

Step 2: Calculate the area of the square

The area of the square is 18 times the area of the triangle, so:

$\text{Area of square} = 18 \times \text{Area of triangle}$

Step 3: Determine the side length of the square

The area of a square is given by:

$\text{Area of square} = (\text{side})^2$

Thus, to find the side length of the square, we solve:

$\text{side} = \sqrt{\text{Area of square}}$

Step 4: Calculate the perimeter of the square

Finally, the perimeter of a square is given by:

$\text{Perimeter of square} = 4 \times \text{side}$

Let me compute these values now.The perimeter of the square is $24 \, \text{cm}$.

Would you like more details on any of the steps? Here are five related questions to further explore the topic:

1. How do you calculate the area of any polygon?
2. How would the solution change if the square's area was a different multiple of the triangle's area?
3. Can you find the length of the diagonal of the square?
4. How would the perimeter change if the triangle had a different base or height?
5. What other types of shapes can be related using their areas?

Tip: Remember that for triangles, the base and height must always be perpendicular to each other when calculating the area.