Two circles of unit radii, are so drawn that the centre of each lies on the circuference of the other. The area of the region , common to both the circles, is :

Options

A. `((4pi - 3sqrt(3)))/(12)`
B. `((4pi-6sqrt(3)))/(12)`
C. `((4pi-3sqrt(3)))/(6)`
D. `((4pi-6sqrt(3)))/(6)`

Answer

To find the area of the region common to two circles of unit radius where the center of each circle lies on the circumference of the other, we can follow these steps:

### Step 1: Understand the Configuration
We have two circles, each with a radius of 1. The center of each circle lies on the circumference of the other circle. Let's denote the centers of the circles as O1 and O2.

### Step 2: Identify Key Points
The points where the circles intersect can be denoted as A and B. The triangle formed by the centers O1, O2, and the intersection points A and B is an equilateral triangle because all sides are equal to the radius of the circles, which is 1.

### Step 3: Calculate the Area of Triangle O1AB
The area of an equilateral triangle can be calculated using the formula:
\[ \text{Area} = \frac{\sqrt{3}}{4} s^2 \]
where \(s\) is the length of a side. Here, \(s = 1\):
\[ \text{Area of triangle O1AB} = \frac{\sqrt{3}}{4} \times 1^2 = \frac{\sqrt{3}}{4} \]

### Step 4: Calculate the Area of the Circular Sectors
Each of the two circles contributes a sector to the common area. The angle subtended by the line segment O1O2 at each center is 60 degrees (since the triangle is equilateral). The area of a sector can be calculated using the formula:
\[ \text{Area of sector} = \frac{\theta}{360} \times \pi r^2 \]
For our case, \(r = 1\) and \(\theta = 60\):
\[ \text{Area of one sector} = \frac{60}{360} \times \pi \times 1^2 = \frac{1}{6} \pi \]
Since there are two such sectors (one from each circle), the total area of the sectors is:
\[ \text{Total area of sectors} = 2 \times \frac{1}{6} \pi = \frac{1}{3} \pi \]

### Step 5: Calculate the Common Area
The common area between the two circles is the total area of the sectors minus the area of the triangle:
\[ \text{Common area} = \text{Total area of sectors} - \text{Area of triangle O1AB} \]
Substituting the values:
\[ \text{Common area} = \frac{1}{3} \pi - \frac{\sqrt{3}}{4} \]

### Step 6: Final Expression
To express the common area in a single fraction, we can find a common denominator:
\[ \text{Common area} = \frac{4\pi}{12} - \frac{3\sqrt{3}}{12} = \frac{4\pi - 3\sqrt{3}}{12} \]

Thus, the area of the region common to both circles is:
\[ \frac{4\pi - 3\sqrt{3}}{12} \]