The area of circle whose radius is 6 cm is trisected by two concentric circles. The radius of the smallest circle is

Options

A. `2sqrt(3) cm`
B. `2sqrt(6) cm`
C. `2 cm`
D. `3 cm`

Answer

To find the radius of the smallest circle that trisects the area of a larger circle with a radius of 6 cm, we can follow these steps:

### Step 1: Calculate the area of the larger circle.
The formula for the area of a circle is given by:
\[ \text{Area} = \pi r^2 \]
where \( r \) is the radius of the circle.

For our larger circle:
\[ r = 6 \text{ cm} \]
So, the area of the larger circle is:
\[ \text{Area} = \pi (6)^2 = \pi \times 36 = 36\pi \text{ cm}^2 \]

### Step 2: Trisect the area of the larger circle.
To trisect the area, we divide the total area by 3:
\[ \text{Area of each section} = \frac{36\pi}{3} = 12\pi \text{ cm}^2 \]

### Step 3: Set up the equation for the area of the smallest circle.
Let the radius of the smallest circle be \( r_1 \). The area of the smallest circle can also be expressed as:
\[ \text{Area} = \pi r_1^2 \]
Setting this equal to the area of one section:
\[ \pi r_1^2 = 12\pi \]

### Step 4: Simplify the equation.
We can divide both sides by \( \pi \) (assuming \( \pi \neq 0 \)):
\[ r_1^2 = 12 \]

### Step 5: Solve for the radius \( r_1 \).
Taking the square root of both sides gives us:
\[ r_1 = \sqrt{12} = \sqrt{4 \times 3} = 2\sqrt{3} \text{ cm} \]

### Conclusion
The radius of the smallest circle is:
\[ \boxed{2\sqrt{3} \text{ cm}} \]