Can I keep BC as the base and radius 6 as the height and calculate the area using formula 1/2 B * H?
You need to have BOTH statements to be able to calculate the area. Without statement 1, we don't know that the radius is necessarily the height. Without statement 2, we don't know that BC is the diameter of the circle and therefore we don't know the base.
Also, note that, to answer the data sufficiency problem, we need only to recognize that both statements combined are sufficient to calculate the area; we don't need to actually calculate the area.
So, in general, on data sufficiency questions, make sure you're not spending time doing extra calculations when you don't need to.
But, yes, given both statements 1 and 2, we could calculate the area using the formula:
Area of triangle = 1/2 * base * height
The base is the diameter = 12
The height is the radius = 6
So we have:
1/2 * 12 * 6 = 36