To determine the volume of one of the six square pyramids formed within a cube, let's break down the problem step by step.
1. Volume of the Cube:
The volume [tex]\( V \)[/tex] of a cube with side length [tex]\( b \)[/tex] is given by:
[tex]\[
V_{\text{cube}} = b^3
\][/tex]
2. Understanding the Pyramids:
Inside the cube, when the four diagonals are drawn, it creates six square pyramids. Each pyramid has its base as one of the square faces of the cube and a height equal to half the body diagonal of the cube.
3. Height of the Pyramids:
For each of these pyramids, the height can be considered as the body diagonal of the cube divided by 2. Given that the height [tex]\( h \)[/tex] provided corresponds with the cube's side, we have:
[tex]\[
h = \frac{\sqrt{3}}{2}b
\][/tex]
4. Volume of a Pyramid:
The volume [tex]\( V \)[/tex] of a pyramid with a square base is given by:
[tex]\[
V_{\text{pyramid}} = \frac{1}{3} \times [\text{Base Area}] \times [\text{Height}]
\][/tex]
The base area of the pyramid is:
[tex]\[
\text{Base Area} = b^2
\][/tex]
Substituting the height [tex]\( h \)[/tex] as:
[tex]\[
V_{\text{pyramid}} = \frac{1}{3} \times b^2 \times 2h
\][/tex]
Simplifying:
[tex]\[
V_{\text{pyramid}} = \frac{1}{3} \times b^2 \times 2 \times \frac{1}{2} = \frac{1}{6} \times b^2 \times b \times 2 = \frac{1}{6}(b^2)(2h)
\][/tex]
5. Comparing the Options:
We need to determine which expression from the options matches our derived formula:
[tex]\[
\frac{1}{6}(b^2)(2h) = \frac{1}{3}b^2h
\][/tex]
We compare it to the options given:
- [tex]\(\frac{1}{6}(b)(b)(2h)\)[/tex] or [tex]\(\frac{1}{3}bh\)[/tex]
- [tex]\(\frac{1}{6}(b)(b)(6h)\)[/tex] or [tex]\(bh\)[/tex]
- [tex]\(\frac{1}{3}(b)(b)(6h)\)[/tex] or [tex]\(2bh\)[/tex]
- [tex]\(\frac{1}{3}(b)(b)(2h)\)[/tex] or [tex]\(\frac{2}{3}bh\)[/tex]
The correct one that matches our derived volume is:
[tex]\[
\frac{1}{6}(b^2)(2h)
\][/tex]
or equivalently:
[tex]\[
\frac{1}{3}b^2h
\][/tex]
Therefore, the correct choice is:
[tex]\[
\frac{1}{6}(b)(b)(2h) \text{ or } \frac{1}{3}bh.
\][/tex]
