Length of a side of a square inscribed in a circle is `asqrt(2)` units. The circumference of the circle is

Options

A. `2 pi a` units
B. `pi a` units
C. `4 pi a` units
D. `(2a)/(pi)` units

Answer

To find the circumference of a circle in which a square is inscribed, we can follow these steps:

### Step 1: Understand the relationship between the square and the circle
The diagonal of the inscribed square is equal to the diameter of the circle. If the side length of the square is given as \( a\sqrt{2} \), we can find the diagonal using the Pythagorean theorem.

### Step 2: Calculate the diagonal of the square
The formula for the diagonal \( d \) of a square with side length \( a \) is:
\[ d = a\sqrt{2} \]
In this case, the side length \( a \) is \( a\sqrt{2} \). Therefore, the diagonal \( d \) is:
\[ d = (a\sqrt{2})\sqrt{2} = a \cdot 2 = 2a \]

### Step 3: Relate the diagonal to the diameter of the circle
Since the diagonal of the square is equal to the diameter \( D \) of the circle, we have:
\[ D = 2a \]

### Step 4: Calculate the radius of the circle
The radius \( R \) of the circle is half of the diameter:
\[ R = \frac{D}{2} = \frac{2a}{2} = a \]

### Step 5: Calculate the circumference of the circle
The formula for the circumference \( C \) of a circle is:
\[ C = 2\pi R \]
Substituting the value of the radius:
\[ C = 2\pi a \]

### Conclusion
Thus, the circumference of the circle is:
\[ \text{Circumference} = 2\pi a \]

---