The diagonal of a square is 'a' units .Then , the area of the square is ________ square units .

Answer

To find the area of a square when the diagonal is given as 'a' units, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Square and its Diagonal**:
- Let's denote the side length of the square as \( x \).
- The diagonal of the square divides it into two right-angled triangles.

2. **Applying the Pythagorean Theorem**:
- According to the Pythagorean theorem, for a right-angled triangle, the square of the hypotenuse (diagonal) is equal to the sum of the squares of the other two sides (which are equal to the sides of the square).
- Therefore, we can write the equation:
\[ a^2 = x^2 + x^2 \]
- This simplifies to:
\[ a^2 = 2x^2 \]

3. **Solving for the Side Length \( x \)**:
- Rearranging the equation gives:
\[ x^2 = \frac{a^2}{2} \]
- Taking the square root of both sides, we find:
\[ x = \frac{a}{\sqrt{2}} \]

4. **Calculating the Area of the Square**:
- The area \( A \) of a square is given by the formula:
\[ A = x^2 \]
- Substituting the value of \( x \):
\[ A = \left(\frac{a}{\sqrt{2}}\right)^2 \]
- This simplifies to:
\[ A = \frac{a^2}{2} \]

5. **Final Answer**:
- Therefore, the area of the square is:
\[ \frac{a^2}{2} \text{ square units} \]