7.5: Volume of Prisms
Learning Objectives
- Find the volume of a prism.
Volume of a Right Rectangular Prism
Volume is a measure of how much space a 3-dimensional figure occupies.
This means that the volume tells you how much a 3-dimensional figure can hold.
What does volume represent?
Volume is the space inside a 3-dimensional solid.
One way to understand volume is to compare it to surface area.
We can use real-life examples of objects to compare volume to surface area:
A fish tank:
Surface area = the glass used to build the outside of the tank
Volume = the water inside the tank
A pillow:
Surface area = the fabric used to make the pillowcase
Volume = the feathers or stuffing inside the pillow
Can you think of other examples?
The basic unit of volume is the cubic unit — cubic centimeter, cubic inch, cubic meter, cubic foot, and so on. Each basic cubic unit has a measure of 1 for its length, width, and height:
______________________ is the measure of space inside a solid object.
The basic unit of volume is a ________________________ unit.
In calculating volume, it is important to know that if 2 polyhedrons (or solids) are congruent, then their volumes are congruent also.
A right rectangular prism is a prism with rectangular bases and the angle between each base and its rectangular lateral sides is also a right angle. You can recognize a right rectangular prism by its “box” shape, like in the diagram below.
The volume of a solid is the sum of the volumes of all of its non-overlapping parts. Using this, we can find the volume of a right rectangular prism by counting boxes.
The box below measures 2 units in height, 4 units in width, and 3 units in depth. Each layer has (\begin{align*}2 \cdot 4\end{align*}) cubes or 8 cubes.
Together, you get 3 groups of (\begin{align*}2 \cdot 4\end{align*}) so the total volume is:
\begin{align*}V & = 2 \cdot 4 \cdot 3\\ &= 24\end{align*}
The volume is 24 cubic units.
This same pattern is true for any right rectangular prism.
Volume is given by the formula:
\begin{align*}\text{Volume}& = \text{length} \cdot \text{width} \cdot \text{height}\\ V & = l \cdot w \cdot h\end{align*}
You can calculate the volume of any right rectangular prism by multiplying the ________________________ of the solid, the ________________________ , and its _________________________ .
Example 1
Find the volume of this prism:
Use the formula for volume of a right rectangular prism:
\begin{align*}V &= l \cdot w \cdot h\\ V &= \underline{\;\;\;\;\;\;\;\;\;\;\;\;\;\;} \cdot \underline{\;\;\;\;\;\;\;\;\;\;\;\;\;\;} \cdot \underline{\;\;\;\;\;\;\;\;\;\;\;\;\;\;}\\ V &= 560\end{align*}
So the volume of this rectangular prism is 560 cubic units.
Reading Check:
1. In your own words, what is volume?
\begin{align*}{\;}\end{align*}
\begin{align*}{\;}\end{align*}
\begin{align*}{\;}\end{align*}
2. True/False: An appropriate unit for the answer to a volume problem is cubic inches.
3. True/False: If 2 solids are congruent, then their volumes are the same.
4. True/False: Volume is calculated by taking the sum of the length, the width, and the height of a solid.
Volume of a Right Prism
Looking at the volume of right prisms with the same height and different bases, you can see a pattern. The computed area of each base is given below. The height of all 3 solids is the same, 10.
Putting the data for each solid into a table, we get:
Solid | Height | Area of base | Volume |
---|---|---|---|
Rectangle | 10 | 300 | 3000 |
Trapezoid | 10 | 140 | 1400 |
Triangle | 10 | 170 | 1700 |
The relationship in each case is clear: when you multiply the height of the solid by the area of its base, you get the volume. This relationship can be proven to establish the following formula for any right prism.
The volume of a right prism is:
\begin{align*}V = Bh\end{align*}
where \begin{align*}B\end{align*} is the area of the base of the 3-dimensional figure and \begin{align*}h\end{align*} is the prism’s height (also called altitude)
To find the volume of a right prism, you ______________________________ the area of its ________________________ by the __________________________ of the prism.
Example 2
Find the volume of the prism with a triangular equilateral base and the dimensions shown in centimeters.
To find the volume, first find the area of the base. In this diagram, the base is actually facing forwards instead of on the bottom. The base is an equilateral triangle as the directions say, so we use the area of a triangle formula:
\begin{align*}A = \frac{1}{2}\ bh\end{align*}
The height (or altitude) of the triangle is 10.38 cm. Each side of the triangle measures 12 cm. So the triangle has the following area:
\begin{align*}A &= \frac{1}{2}\ bh\\ &= \frac{1}{2} (12)(10.38)\\ &= 62.28 \ cm^2\end{align*}
Now use the formula for the volume of the prism, \begin{align*}V = Bh\end{align*}, where \begin{align*}B\end{align*} is the area of the base (the area of the triangle) and \begin{align*}h\end{align*} is the height of the prism.
Remember that the "height" of the prism is the distance between the bases, so in this case the height of the prism is 15 cm. Imagine that the prism is lying on its side.
\begin{align*}V& = Bh\\ &= (62.28)(15)\\ &= 934.2\end{align*}
Thus, the volume of the prism is \begin{align*}934.2 \ cm^3\end{align*} (or cubic centimeters).
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