7.6: Volume of Pyramids
Learning Objectives
- Find the volume of a pyramid.
Estimate the Volume of a Pyramid and Prism
Which has a greater volume, a prism or a pyramid, if both solids have the same base and height? To find out, compare prisms and pyramids that have congruent bases and the same height.
On the left is a base for a triangular prism and a triangular pyramid. Both figures have the same height. Compare the two figures. Which one appears to have a greater volume?
The prism may appear to be greater in volume, but how can you prove that the volume of the prism is greater than the volume of the pyramid? You can put one figure inside of the other. The figure that is smaller will fit inside of the other figure.
This is shown in the diagram above. Both figures have congruent bases and the same height, so when you put the pyramid inside the prism, their bases overlap exactly. Since the shapes are the same height, the top of the pyramid touches the top of the prism.
The pyramid clearly fits inside of the prism, so the volume of the pyramid must be smaller.
Given a prism and a pyramid with congruent bases and the same height, which has a bigger volume? ________________________________
Example 1
Show that the volume of a square prism is greater than the volume of a square pyramid.
Draw or make a square prism and a square pyramid that have congruent bases and the same height.
Now place one figure inside of the other. The pyramid fits inside of the prism:
When two figures have the same height and the same base, the volume of the prism is greater.
In general, when you compare 2 figures that have congruent bases and are equal in height, the prism will have a greater volume than the pyramid.
The reason should be obvious. At the “bottom,” both figures start out the same — with a square base. But the pyramid quickly slants inward, “cutting away” large amounts of material while the prism does not slant.
Reading Check:
1. True/False: Prisms and pyramids with congruent bases have the same volume.
2. True/False: A pyramid will fit inside a prism with the same base and height.
Find the Volume of a Pyramid and Prism
Given the figure on the previous page, in which a square pyramid is placed inside of a square prism, we now ask: how many of these pyramids would fit inside of the prism?
To find out, obtain a square prism and square pyramid that are both hollow, both have no bottom, and both have the same height and congruent bases.
Now turn the figures upside down. Fill the pyramid with liquid. How many full pyramids of liquid will fill the prism up to the top?
In fact, it takes exactly 3 full pyramids to fill the prism. Since the volume of the prism is:
\begin{align*}V = Bh\end{align*}
where \begin{align*}B\end{align*} stands for the area of the base and \begin{align*}h\end{align*} is the height of the prism, we can write:
\begin{align*}&3 \cdot \text{(volume of a square pyramid)} = \text{volume of a square prism}\\ &\qquad \qquad \quad \qquad \qquad \quad \qquad \text{or}:\\ &\text{volume of a square pyramid} = \frac{1}{3} \ \text{(volume of a square prism)}\end{align*}
And, since the volume of a square prism is \begin{align*}Bh\end{align*}, the volume of a square pyramid is:
\begin{align*}V = \frac{1}{3} Bh\end{align*}
Volume of a Pyramid
Given a right pyramid with a base that has area \begin{align*}B\end{align*} and height \begin{align*}h\end{align*}:
\begin{align*}V = \frac{1}{3}\ Bh\end{align*}
Example 2
Find the volume of a pyramid with a right triangle base with sides that measure 5 cm, 12 cm, and 13 cm. The height of the pyramid is 15 cm.
First find the area of the base:
Since the base is a right triangle and the 3 sides measure 5 cm, 12 cm, and 13 cm, the longest side (13) must be the hypotenuse. The 2 shorter sides (5 and 12) are the legs of the right triangle. Use the leg lengths as the base and height of the triangle:
\begin{align*}A &= \frac{1}{2}\ bh\\ &= \frac{1}{2} (5)(12)\\ & = 30 \ \text{square cm}\end{align*}
Now use the formula above for the volume of a pyramid:
\begin{align*}V &= \frac{1}{3}\ Bh\\ &= \frac{1}{3} (30)(15)\\ &= 150 \ \text{cubic cm}\end{align*}
Reading Check:
1. True/False: Since the volume of a prism is 3 times the volume of a pyramid, the volume of a pyramid is half the volume of a prism.
2. Fill in the blanks: In the formula for the volume of a right pyramid, \begin{align*}V = \frac{1}{3}\ Bh\end{align*},
\begin{align*}V\end{align*} stands for the _________________________,
\begin{align*}B\end{align*} stands for the ______________________ of the _____________________ and
\begin{align*}h\end{align*} stands for the _________________________ of the pyramid.
3. If you were to fill a pyramid with liquid and pour that liquid into a prism with the same base as the pyramid, how many full pyramids would fit into the prism?
\begin{align*}{\;}\end{align*}
\begin{align*}{\;}\end{align*}
\begin{align*}{\;}\end{align*}
Graphic Organizer for Unit 7
Shape and Picture | ||||
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Surface Area | Volume | |||
Formula | What do I need to know? | Formula | What do I need to know? | |
Prism | ||||
Pyramid |
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