Correct Answer: (C)

1. Analysis of the Question: The area of a circumcircle is given by $A = \pi R^2$, where $R$ is the circumradius. For any triangle with sides $a, b, c$, the circumradius is $R = \frac{abc}{4\Delta}$, where $\Delta$ is the area of the triangle.

2. Evaluating Statement (1): The ratio 3 : 4 : 5 indicates that the triangle is a right-angled triangle. In a right-angled triangle, the circumradius $R$ is half of the hypotenuse. However, without absolute side lengths, we only know the ratio of $R$ to the sides, not its numerical value. Thus, (1) alone is not sufficient.

3. Evaluating Statement (2): Knowing only the perimeter is 36 cm does not define the shape or type of the triangle. A perimeter of 36 cm could belong to an equilateral triangle, an isosceles triangle, or infinitely many others, each having a different circumradius. Thus, (2) alone is not sufficient.

4. Combining Both Statements: From (1), we know the sides are $3k, 4k, 5k$. From (2), the perimeter is $3k + 4k + 5k = 36$, which gives $12k = 36$, so $k = 3$. The sides are therefore 9 cm, 12 cm, and 15 cm. Since it is a right-angled triangle, the hypotenuse is 15 cm, and the circumradius $R = \frac{15}{2} = 7.5$ cm. The area of the circumcircle can now be uniquely calculated as $\pi (7.5)^2$.

Test Prep Tip: In Data Sufficiency for Geometry, remember that finding a "ratio" provides the shape, while a "linear measurement" (like perimeter or a single side) provides the scale. You typically need both to find a specific area or volume. Use the property that for right-angled triangles, the circumcenter lies at the midpoint of the hypotenuse.