A square ABCD is inscribed in a circle of unit radius. Semicircles are described on each side as a diameter. The area of the region bounded by the four semicircles and the circle is

Options

A. 1 sq. unit
B. 2 sq. unit
C. 1.5 sq. unit
D. 2.5 sq. unit

Answer

To solve the problem step by step, we need to find the area of the region bounded by the four semicircles and the circle.

### Step 1: Understand the Configuration
We have a square ABCD inscribed in a circle with a unit radius. The semicircles are drawn on each side of the square as diameters.

### Step 2: Find the Diameter of the Circle
Since the radius of the circle is 1 unit, the diameter \(D\) of the circle is:
\[ D = 2 \times \text{radius} = 2 \times 1 = 2 \text{ units} \]

### Step 3: Relate the Diameter to the Square
The diameter of the circle is equal to the diagonal of the square. Let \(s\) be the side length of the square. The diagonal \(d\) of the square can be expressed as:
\[ d = s\sqrt{2} \]
Setting this equal to the diameter of the circle:
\[ s\sqrt{2} = 2 \]
Solving for \(s\):
\[ s = \frac{2}{\sqrt{2}} = \sqrt{2} \text{ units} \]

### Step 4: Find the Radius of the Semicircles
The diameter of each semicircle is equal to the side of the square, which is \(s = \sqrt{2}\). Therefore, the radius \(r\) of each semicircle is:
\[ r = \frac{s}{2} = \frac{\sqrt{2}}{2} \text{ units} \]

### Step 5: Calculate the Area of One Semicircle
The area \(A\) of one semicircle is given by the formula:
\[ A = \frac{1}{2} \pi r^2 \]
Substituting the radius:
\[ A = \frac{1}{2} \pi \left(\frac{\sqrt{2}}{2}\right)^2 = \frac{1}{2} \pi \left(\frac{2}{4}\right) = \frac{1}{2} \pi \left(\frac{1}{2}\right) = \frac{\pi}{4} \text{ square units} \]

### Step 6: Calculate the Total Area of Four Semicircles
Since there are four semicircles:
\[ \text{Total Area of Semicircles} = 4 \times \frac{\pi}{4} = \pi \text{ square units} \]

### Step 7: Calculate the Area of the Circle
The area of the circle with a radius of 1 unit is:
\[ \text{Area of Circle} = \pi r^2 = \pi (1)^2 = \pi \text{ square units} \]

### Step 8: Calculate the Area of the Square
The area of the square with side length \(s = \sqrt{2}\) is:
\[ \text{Area of Square} = s^2 = (\sqrt{2})^2 = 2 \text{ square units} \]

### Step 9: Find the Area of the Shaded Region
The area of the region bounded by the semicircles and the circle is given by:
\[ \text{Area of Shaded Region} = \text{Total Area of Semicircles} - (\text{Area of Circle} - \text{Area of Square}) \]
Substituting the values:
\[ \text{Area of Shaded Region} = \pi - (\pi - 2) = 2 \text{ square units} \]

### Final Answer
Thus, the area of the region bounded by the four semicircles and the circle is:
\[ \boxed{2} \text{ square units} \]