A rectangular sheet of paper is 3 cm × 2 cm in area. If the greatest possible circle is cut out from it, the area of the remaining paper is

Options

A. `(9 - pi)` sq.cm
B. `(9 - 4 pi)` sq.cm
C. `(6 - pi)` sq.cm
D. `(6 - 4 pi)` sq.cm

Answer

To solve the problem step by step, we will find the area of the rectangular sheet of paper, determine the radius of the largest circle that can be cut from it, calculate the area of that circle, and then find the area of the remaining paper.

### Step 1: Calculate the area of the rectangular sheet of paper.
The dimensions of the rectangular sheet are given as 3 cm and 2 cm.

\[ \text{Area of rectangle} = \text{Length} \times \text{Width} = 3 \, \text{cm} \times 2 \, \text{cm} = 6 \, \text{cm}^2 \]

### Step 2: Determine the radius of the largest circle that can be cut from the rectangle.
The largest circle that can fit inside the rectangle will have a diameter equal to the smaller dimension of the rectangle. In this case, the smaller dimension is 2 cm.

\[ \text{Diameter of circle} = 2 \, \text{cm} \implies \text{Radius} = \frac{\text{Diameter}}{2} = \frac{2 \, \text{cm}}{2} = 1 \, \text{cm} \]

### Step 3: Calculate the area of the circle.
The area of a circle is given by the formula:

\[ \text{Area of circle} = \pi r^2 \]

Substituting the radius we found:

\[ \text{Area of circle} = \pi (1 \, \text{cm})^2 = \pi \, \text{cm}^2 \]

### Step 4: Calculate the area of the remaining paper.
To find the area of the remaining paper after cutting out the circle, we subtract the area of the circle from the area of the rectangle:

\[ \text{Area of remaining paper} = \text{Area of rectangle} - \text{Area of circle} \]

Substituting the values we calculated:

\[ \text{Area of remaining paper} = 6 \, \text{cm}^2 - \pi \, \text{cm}^2 \]

Thus, the area of the remaining paper is:

\[ \text{Area of remaining paper} = 6 - \pi \, \text{cm}^2 \]

### Final Answer:
The area of the remaining paper is \( 6 - \pi \, \text{cm}^2 \).

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