The area of a rectangle is 216 meters square and the ratio of length to breadth of the rectangle is 3 : 2. If diameter of a circle is $1\frac{1}{6}\text{th}$ of the breadth of the rectangle, then find the difference between area of the rectangle and circle (in meters square)?

To find the difference between the area of the rectangle and the circle, let us solve the problem step-by-step.

Step 1: Find the dimensions of the rectangle.
Let the length of the rectangle be 3x meters and the breadth of the rectangle be 2x meters, since the ratio of length to breadth is 3 : 2.
The area of a rectangle is given by the formula:
Area = length × breadth
Given that the area of the rectangle is 216 m²:
$3x\times 2x=216$
$6x^2=216$
$x^2=\frac{216}{6}$
$x^2=36$
$x=6$
Therefore, the breadth of the rectangle is:
Breadth = 2x = 2 × 6 = 12 meters.

Step 2: Find the diameter and radius of the circle.
The diameter of the circle is $1\frac{1}{6}\text{th}$ of the breadth of the rectangle.
Converting the mixed fraction $1\frac{1}{6}$ to an improper fraction, we get:
$1\frac{1}{6}=\frac{1\times 6+1}{6}=\frac{7}{6}$
So, the diameter of the circle is:
$\text{Diameter}=\frac{7}{6}\times 12=14\text{meters}$
The radius (r) of the circle is half of the diameter:
$r=\frac{14}{2}=7\text{meters}$

Step 3: Calculate the area of the circle.
The area of a circle is calculated using the formula $\pi r^2$ (taking $\pi \approx \frac{22}{7}$):
$\text{Area of the circle}=\frac{22}{7}\times 7\times 7=22\times 7=154{\text{m}}^2$

Step 4: Find the difference between the two areas.
Difference = Area of rectangle - Area of circle
$\text{Difference}=216-154=62{\text{m}}^2$

The difference between the area of the rectangle and the circle is 62 meters square, which corresponds to Option 62.