To calculate the height of the triangular sandwich, we can use the formula for the area of a triangle:
\[ \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} \]
Let \( A \) be the area, \( b \) be the base, and \( h \) be the height. Plugging in the values we have:
\[ 7 = \frac{1}{2} \times 4 \times h \]
First, simplify the right side:
\[ 7 = 2 \times h \]
Now, to solve for \( h \), divide both sides by 2:
\[ h = \frac{7}{2} = 3.5 \]
Thus, the height of the sandwich is \( \boxed{3.5} \) inches.