What is the area of the largest semicircle that can be inscribed in the square whose diagonal is `asqrt2` units?

Options

A. `0.12pia^2`
B. `0.125pia^2`
C. `0.22pia^2`
D. `1.25pia^2`

Answer

To find the area of the largest semicircle that can be inscribed in a square whose diagonal is \( a\sqrt{2} \) units, we can follow these steps:

### Step 1: Determine the side length of the square
The diagonal \( d \) of a square is related to its side length \( a \) by the formula:
\[ d = a\sqrt{2} \]
Given that the diagonal of the square is \( a\sqrt{2} \), we can conclude that the side length \( a \) of the square is:
\[ \text{Side length} = a \]

### Step 2: Identify the diameter of the semicircle
The largest semicircle that can be inscribed in the square will have its diameter equal to the side length of the square. Therefore, the diameter \( D \) of the semicircle is:
\[ D = a \]

### Step 3: Calculate the radius of the semicircle
The radius \( r \) of the semicircle is half of the diameter:
\[ r = \frac{D}{2} = \frac{a}{2} \]

### Step 4: Calculate the area of the semicircle
The area \( A \) of a semicircle is given by the formula:
\[ A = \frac{1}{2} \pi r^2 \]
Substituting the value of \( r \):
\[ A = \frac{1}{2} \pi \left(\frac{a}{2}\right)^2 \]
Calculating \( \left(\frac{a}{2}\right)^2 \):
\[ \left(\frac{a}{2}\right)^2 = \frac{a^2}{4} \]
Now substituting this back into the area formula:
\[ A = \frac{1}{2} \pi \cdot \frac{a^2}{4} = \frac{\pi a^2}{8} \]

### Step 5: Express the area in decimal form
To express the area in decimal form:
\[ A = 0.125 \pi a^2 \]

### Conclusion
Thus, the area of the largest semicircle that can be inscribed in the square is:
\[ \boxed{0.125 \pi a^2} \]