Is the area of the largest circle that can be drawn inside a rectangle of length `a` `cm` and breadth `b` `cm` `(a gt b)` is `pi b^(2)` `cm^2` ? Why ?

Explanation

To determine if the area of the largest circle that can be drawn inside a rectangle of length `a` cm and breadth `b` cm (where `a > b`) is equal to `πb²` cm², we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Dimensions**:
- Given a rectangle with length `a` cm and breadth `b` cm, where `a > b`.

2. **Identifying the Largest Circle**:
- The largest circle that can fit inside the rectangle will have its diameter equal to the smaller dimension of the rectangle, which is the breadth `b`. This is because if we tried to use the length `a` as the diameter, the circle would extend beyond the rectangle.

3. **Calculating the Radius**:
- The diameter of the circle is `b`, so the radius `r` of the circle will be:
\[ r = \frac{b}{2} \text{ cm} \]

4. **Finding the Area of the Circle**:
- The area `A` of a circle is given by the formula:
\[ A = \pi r^2 \]
- Substituting the radius we found:
\[ A = \pi \left(\frac{b}{2}\right)^2 = \pi \cdot \frac{b^2}{4} = \frac{\pi b^2}{4} \text{ cm}^2 \]

5. **Comparing with the Given Statement**:
- The question states that the area of the largest circle is `πb²` cm². However, we calculated the area to be:
\[ \frac{\pi b^2}{4} \text{ cm}^2 \]
- Therefore, the statement in the question is incorrect.

6. **Conclusion**:
- The area of the largest circle that can be drawn inside the rectangle is not `πb²` cm², but rather `\(\frac{\pi b^2}{4}\)` cm².

### Final Answer:
No, the area of the largest circle that can be drawn inside a rectangle of length `a` cm and breadth `b` cm (where `a > b`) is not `πb²` cm²; it is actually `\(\frac{\pi b^2}{4}\)` cm².