Certainly! Let's solve this step-by-step to find the volume of the pyramid.
1. Identify the given information:
- The pyramid has a square base with side length [tex]\( s \)[/tex].
- The height of the pyramid is [tex]\( \frac{2}{3} \)[/tex] of its side, which gives the height [tex]\( h = \frac{2}{3}s \)[/tex].
2. Recall the formula for the volume of a pyramid:
The volume [tex]\( V \)[/tex] of a pyramid is given by:
[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 [tex]\( s \)[/tex], so the base area [tex]\( A \)[/tex] is:
[tex]\[
A = s^2
\][/tex]
4. Substitute the base area and height into the volume formula:
We have:
[tex]\[
V = \frac{1}{3} \times s^2 \times \frac{2}{3}s
\][/tex]
5. Simplify the expression:
Combining the terms, we get:
[tex]\[
V = \frac{1}{3} \times s^2 \times \frac{2}{3}s
\][/tex]
[tex]\[
V = \frac{1}{3} \times \frac{2}{3}s^3
\][/tex]
[tex]\[
V = \frac{2}{9}s^3
\][/tex]
6. Simplify even further if needed:
We observe that [tex]\(\frac{2}{9}\)[/tex] can be simplified as a decimal:
[tex]\[
V \approx 0.222222222222222s^3
\][/tex]
Comparing this result with the options given, we see that none are an exact match in their form provided, but the numerical coefficient can guide us. The closest and correct form matching our calculations is:
```
None of the given options precisely match our simplified form. Thus, it appears there's a discrepancy or error in the question options provided, considering our correct derivation.
```
However, in the usual multiple-choice scenario, if this were simplified further and options adapted accordingly, the true answer aligns accurately with our calculation surpassing provided choices.
Therefore, the correct expression for the volume of the pyramid is:
[tex]\[ \boxed{0.222222222222222s^3} \][/tex]
