To determine the correct formula for the volume of a pyramid with a square base and a height equal to half the length of the side, we can follow these steps:
1. Identify the parameters:
- Let [tex]\( s \)[/tex] be the side length of the square base.
- The height of the pyramid is given as [tex]\( \frac{1}{2} \)[/tex] of the side length, so height [tex]\( h = \frac{s}{2} \)[/tex].
2. Recall the formula for the volume of a pyramid:
[tex]\[
V = \frac{1}{3} \times (\text{Base Area}) \times (\text{Height})
\][/tex]
3. Calculate the base area:
- The base is a square with side length [tex]\( s \)[/tex], so the area of the base [tex]\( B \)[/tex] is:
[tex]\[
B = s^2
\][/tex]
4. Substitute the base area and height into the volume formula:
[tex]\[
V = \frac{1}{3} \times s^2 \times \frac{s}{2}
\][/tex]
5. Simplify the expression:
[tex]\[
V = \frac{1}{3} \times s^2 \times \frac{s}{2} = \frac{1}{3} \times \frac{1}{2} \times s^3 = \frac{1}{6} s^3
\][/tex]
Therefore, the correct formula for the volume of the pyramid is:
[tex]\[
V = \frac{1}{6} s^3
\][/tex]
Thus, the correct answer is [tex]\( \boxed{B} \)[/tex].
