Question

deriving a formula for volume of a pyramid
the area of the base of the cube, ( b ), is ((b)(b)) square units.
the volume of the cube is (\boldsymbol{\text{dropdown}}) cubic units.
the height of each pyrami(\boldsymbol{\text{dropdown}}). therefore, ( b = 2h ).
there are (\boldsymbol{\text{dropdown}}) square pyra(\boldsymbol{\text{dropdown}}) e same base and height that exactly fill (\boldsymbol{\text{dropdown}}) pe.
therefore, the volume of one pyramid is (\boldsymbol{\text{dropdown}}) or (\frac{1}{3}bh).

Explanation:

Step1: Volume of the cube

The volume of a cube with side length \( b \) is given by the formula \( V = \text{side}^3 \). So, substituting the side length \( b \), we get the volume of the cube as \( (b)(b)(b) \) or \( b^3 \).

Step2: Height of each pyramid

From the diagram, we know that the edge length of the cube \( b \) is related to the height \( h \) of the pyramid by \( b = 2h \), so the height of each pyramid is \( \frac{b}{2} \).

Step3: Number of pyramids filling the cube

Looking at the cube divided into pyramids, we can see that there are 6 square pyramids with the same base and height that exactly fill the cube. This is because the cube can be divided into 6 congruent pyramids (by connecting the center of the cube to the vertices).

Step4: Volume of one pyramid

The volume of the cube is \( b^3 \) (or \( (b)(b)(b) \)). Since 6 pyramids fill the cube, the volume of one pyramid is \( \frac{\text{Volume of cube}}{6} \). Substituting the volume of the cube, we get \( \frac{(b)(b)(b)}{6} \) or simplifying, since \( B = b^2 \) (the base area) and \( h=\frac{b}{2} \), we can also see that \( \frac{1}{3}Bh=\frac{1}{3}b^2\times\frac{b}{2}=\frac{b^3}{6} \), which matches \( \frac{(b)(b)(b)}{6} \).

Answer:

s (filling in the blanks):

• The volume of the cube is \(\boldsymbol{(b)(b)(b)}\) (or \(b^3\)) cubic units.
• The height of each pyramid: From \( b = 2h \), \( h=\boldsymbol{\frac{b}{2}} \).
• There are \(\boldsymbol{6}\) square pyramids with the same base and height that exactly fill the cube.
• The volume of one pyramid is \(\boldsymbol{\frac{(b)(b)(b)}{6}}\) (or \(\frac{1}{6}b^3\)) or \(\frac{1}{3}Bh\).