A figure consists of a square of side 'a' m with semicircle drawn on the outside of the square. The area ( in `m^2`) of the figure so formed will be :

Options

A. `a^(2) (pi + 1)`
B. ` a^(2)(pi + 1/4)`
C. `a^(2) + (pi a^(2))/(2)`
D. none of these

Answer

To find the area of the figure consisting of a square with a semicircle drawn on the outside, we can follow these steps:

### Step 1: Calculate the area of the square
The area \( A_s \) of a square with side length \( a \) is given by the formula:
\[ A_s = a^2 \]

### Step 2: Determine the radius of the semicircle
Since the semicircle is drawn on the outside of the square, its diameter is equal to the side of the square. Therefore, the radius \( r \) of the semicircle is:
\[ r = \frac{a}{2} \]

### Step 3: Calculate the area of the semicircle
The area \( A_{sc} \) of a semicircle is given by the formula:
\[ A_{sc} = \frac{1}{2} \pi r^2 \]
Substituting the value of \( r \):
\[ A_{sc} = \frac{1}{2} \pi \left(\frac{a}{2}\right)^2 = \frac{1}{2} \pi \left(\frac{a^2}{4}\right) = \frac{\pi a^2}{8} \]

### Step 4: Calculate the total area of the figure
The total area \( A_t \) of the figure is the sum of the area of the square and the area of the semicircle:
\[ A_t = A_s + A_{sc} = a^2 + \frac{\pi a^2}{8} \]

### Step 5: Factor out \( a^2 \)
We can factor \( a^2 \) out of the total area:
\[ A_t = a^2 \left(1 + \frac{\pi}{8}\right) \]

### Final Result
Thus, the area of the figure is:
\[ A_t = a^2 \left(1 + \frac{\pi}{8}\right) \text{ m}^2 \]