Find the area of a square, the length of whose diagonal is 64 cm.

Answer

To find the area of a square when the length of its diagonal is given, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the relationship between the diagonal and the side of the square**:
The diagonal \( d \) of a square is related to its side length \( s \) by the formula:
\[ d = s\sqrt{2} \]
This means that the side length can be expressed in terms of the diagonal:
\[ s = \frac{d}{\sqrt{2}} \]

2. **Substitute the given diagonal length**:
Given that the diagonal \( d = 64 \) cm, we can substitute this value into the formula for the side length:
\[ s = \frac{64}{\sqrt{2}} \]

3. **Simplify the expression for side length**:
To simplify \( \frac{64}{\sqrt{2}} \), we can multiply the numerator and the denominator by \( \sqrt{2} \):
\[ s = \frac{64 \sqrt{2}}{2} = 32\sqrt{2} \text{ cm} \]

4. **Calculate the area of the square**:
The area \( A \) of a square is given by the formula:
\[ A = s^2 \]
Substituting the value of \( s \):
\[ A = (32\sqrt{2})^2 \]
\[ A = 32^2 \cdot (\sqrt{2})^2 \]
\[ A = 1024 \cdot 2 = 2048 \text{ cm}^2 \]

5. **Final answer**:
Therefore, the area of the square is:
\[ \text{Area} = 2048 \text{ cm}^2 \]