What is the area of a square with diagonal of 6?

Answer

To find the area of a square with a diagonal of 6, we can follow these steps:

### Step 1: Understand the relationship between the side length and the diagonal of a square.
The diagonal \( d \) of a square is related to the side length \( a \) by the formula:
\[ d = a \sqrt{2} \]

### Step 2: Set up the equation using the given diagonal.
Given that the diagonal \( d \) is 6, we can write:
\[ a \sqrt{2} = 6 \]

### Step 3: Solve for the side length \( a \).
To find \( a \), divide both sides of the equation by \( \sqrt{2} \):
\[ a = \frac{6}{\sqrt{2}} \]

### Step 4: Simplify the expression for \( a \).
To simplify \( \frac{6}{\sqrt{2}} \), we can multiply the numerator and the denominator by \( \sqrt{2} \):
\[ a = \frac{6 \sqrt{2}}{2} = 3\sqrt{2} \]

### Step 5: Calculate the area of the square.
The area \( A \) of a square is given by the formula:
\[ A = a^2 \]
Substituting \( a = 3\sqrt{2} \):
\[ A = (3\sqrt{2})^2 = 9 \cdot 2 = 18 \]

### Conclusion
Thus, the area of the square with a diagonal of 6 is:
\[ \boxed{18} \]

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