To rearrange the surface area equation \( S = \frac{1}{2}lp + B \) for the quantity of interest \( B \), follow these steps:
1. Start with the original equation:
\[
S = \frac{1}{2}lp + B
\]
2. Subtract \( \frac{1}{2}lp \) from both sides:
\[
S - \frac{1}{2}lp = B
\]
3. Therefore, the rearranged equation is:
\[
B = S - \frac{1}{2}lp
\]
So, the final answer is:
\[
B = S - \frac{1}{2}lp
\]