jmc

If any linear dimension (such as radius, side length, height, etc.) of a closed, two-dimensional figure is multiplied by $k$ while the shape of the figure remains the same, the area of the figure is multiplied by $k^2$. Since the area is multiplied by 4 in going from the smaller triangle to the larger triangle, we have $k^2=4$ which implies $k=2$. Therefore, each linear dimension is multiplied by 2, so the height of the larger triangle is $(3\text{ cm})\times 2=\boxed{6}$ centimeters.