To find the height of the triangular sandwich, we can use the formula for the area of a triangle:
\[ \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} \]
In this case, we know the area is 7 square inches and the base measures 4 inches. Plugging in the known values into the formula gives us:
\[ 7 = \frac{1}{2} \times 4 \times \text{height} \]
Now, simplify the equation:
\[ 7 = 2 \times \text{height} \]
To solve for the height, divide both sides by 2:
\[ \text{height} = \frac{7}{2} = 3.5 \text{ inches} \]
Thus, the height of the sandwich is 3.5 inches.