Expressions, Equations, and Relationships>Explaining Volume Relationships Between Triangular Prisms and Pyramids(TEKS.Math.7.8.B)
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Texas 7th Grade Math › Expressions, Equations, and Relationships>Explaining Volume Relationships Between Triangular Prisms and Pyramids(TEKS.Math.7.8.B)
A triangular prism has base area 30 cm² and height 12 cm, giving volume 360 cm³. A triangular pyramid with the same base and the same height has volume 120 cm³.
What is the relationship between these volumes when the prism and pyramid share congruent bases and heights?
They are always equal if the base and height match.
The pyramid's volume is 1/3 of the prism's volume.
The pyramid's volume is 3 times the prism's volume.
The prism uses $V=\frac{1}{3}Bh$ while the pyramid uses $V=Bh$.
Explanation
For congruent bases and heights, $V_{\text{prism}}=Bh$ and $V_{\text{pyramid}}=\tfrac{1}{3}Bh$, so the pyramid has $\tfrac{1}{3}$ the volume of the prism. Geometrically, three identical pyramids with the same base and height can fill the prism exactly.
Consider the formulas for solids with the same triangular base area $B$ and height $h$: prism volume $V=Bh$ and triangular pyramid volume $V=\frac{1}{3}Bh$.
How do the formulas $V=Bh$ (prism) and $V=\frac{1}{3}Bh$ (pyramid) show the relationship between their volumes when bases and heights match?
They show the pyramid has 3 times the prism's volume because $\frac{1}{3}Bh$ is larger than $Bh$.
They show $B$ must be tripled for the pyramid to match the prism.
They show the volumes can't be compared without slant height.
They show the pyramid's volume is $\frac{1}{3}$ of the prism's volume for the same $B$ and $h$.
Explanation
With the same $B$ and $h$, $V_{\text{prism}}=Bh$ and $V_{\text{pyramid}}=\tfrac{1}{3}Bh$, so $V_{\text{pyramid}}=\tfrac{1}{3}V_{\text{prism}}$. Geometrically, three such pyramids fit exactly into the prism.
A triangular prism has base area 18 in² and height 10 in. A triangular pyramid has the same base and height.
Which equation correctly relates the volumes of the pyramid and the prism in this situation?
$V_{\text{pyramid}}=\frac{1}{3}V_{\text{prism}}$
$V_{\text{pyramid}}=V_{\text{prism}}$
$V_{\text{prism}}=\frac{1}{3}V_{\text{pyramid}}$
$V_{\text{pyramid}}=\frac{1}{3}B$
Explanation
Using $V_{\text{prism}}=Bh$ and $V_{\text{pyramid}}=\tfrac{1}{3}Bh$ with the same $B$ and $h$, we get $V_{\text{pyramid}}=\tfrac{1}{3}V_{\text{prism}}$. This matches the geometric idea that three such pyramids fill the prism.
A triangular pyramid has volume 50 m³. A triangular prism with a congruent base and the same height is made from the same base and height.
What is the prism's volume, and what relationship explains your answer?
50 m³, because the volumes are equal when base and height match
16.7 m³, because $V=\frac{1}{3}Bh$ for prisms
150 m³, because a prism with the same base and height has 3 times the pyramid's volume
100 m³, because the base area is doubled for prisms
Explanation
For the same $B$ and $h$, $V_{\text{prism}}=Bh$ and $V_{\text{pyramid}}=\tfrac{1}{3}Bh$, so $V_{\text{prism}}=3,V_{\text{pyramid}}=3\times 50=150$ m³. Three congruent pyramids fill the prism.
A triangular prism with base area 24 m² and height 9 m has volume 216 m³. A triangular pyramid with the same base and height has volume 72 m³.
Which statement best explains this relationship using the formulas?
Since $V=Bh$ for both, the volumes must be equal when the base and height match.
Because $V_{\text{prism}}=Bh$ and $V_{\text{pyramid}}=\frac{1}{3}Bh$, the pyramid's volume is one-third of the prism's, so $72=\frac{1}{3}\cdot 216$.
The prism's volume is $\frac{1}{3}$ of the pyramid's because the base area is triangular.
The pyramid uses the wrong base area; it should use perimeter instead of area.
Explanation
With congruent base area $B$ and height $h$, $V_{\text{prism}}=Bh$ while $V_{\text{pyramid}}=\tfrac{1}{3}Bh$. Thus $V_{\text{pyramid}}=\tfrac{1}{3}V_{\text{prism}}$, matching $216$ m³ and $72$ m³. Geometrically, three such pyramids fill the prism.