Chapter 45: Volumes
It can be fun to introduce students to the computation of volumes. It will work best if we
have enough supplies on hand for all the kids, but the supplies are all quite simple: cube blocks,
cans, rice, paper, tape, rulers, protractors, and scissors.
The best way to begin the study of volumes is with a very large quantity of solid cubes.
The wooden alphabet blocks that are frequently found in playrooms will work excellently – the
more, the merrier. Try to make sure that all the cubes are the same size, however, so separate
them by size and just use the size of which you have the most blocks.
Explain to the class that volume is the amount of space that a solid object takes up. Just
as we used squares to measure the area, we use cubes to measure volume. To reinforce this idea,
you can build several small objects with some of the blocks, then challenge the class to put them
in order from largest volume to smallest volume. As a very simple example, consider the
following three arrangements of blocks:
The middle arrangement is the largest, with a volume of 6 cubes, the last arrangement has a
volume of 5 blocks, and the first arrangement is the smallest, with a volume of 4 blocks.
It should not take too many exercises like these before your class is easily able to count
the volume of a building made of blocks. If they really want to play, you can divide the class up
into groups and give each one a set of 20 blocks. Tell each group to hide some number of their
blocks, and then use the rest to build a structure. When everything is built, each kid takes a piece
of paper and then walks around the room, looking at each structure and trying to guess its
volume. When everyone has written down a guess for each structure, then the class sits down.
Each group, in turn, then explains the volume of their structure. When the kids walk around,
they are not allowed to touch the buildings. However, for the final demonstration, the groups
can take apart the buildings to make the number of blocks clear. If you want to make the game
competative, you can have each team score a point for every student in the class (other than
themselves) who guesses wrong on their volume. This will encourage the groups to be creative
and tricky in putting together their buildings.
It should not take too much time with this game for your students to understand that
volumes are measured in cubes. If they really enjoy the game, you can do it as a warm-up at the
beginning of a lesson on volume.
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When the students have this basic grasp on volume, we can move on to calculate the
volumes of boxes. To begin with, have the students build a one-level arrangement of blocks in
the shape of a rectangle. This can be done individually, in small groups, or as a class – largely
dictated by the amount of blocks and table surface available. Some of the possible arrangements
might be:
Have your students discuss the volume of each structure. Hopefully, at least one student will
notice that the blocks need not be counted individually, but that it is enough to multiply the
length and the width, just as we would to find the area of the rectangle. For example, the first
arrangement above has 2 × 3 = 6 blocks in volume, the second has 3 × 3 = 9 blocks of volume,
and the third has 2 × 5 = 10 blocks in volume. If no student points this out so clearly, then make
a really big rectangle (something like a 6 by 9) and challenge the students to see who can find the
volume first. It is very likely that the first student will have multiplied to find the answer. Ask
the student to explain his or her method, and use leading questions until the answer comes out.
You could play the small-groups game again, and have each group build a big rectangle
of blocks. This time, however, you might find that every single student guesses all the volumes
correctly. There are no opportunities for hiding blocks and leaving gaps when the arrangements
have to be solid rectangles like the above. While the game is less fun, it should indicate that
your students are well along the path to understanding volume!
Next, have students build arrangements of blocks that are rectangles when viewed from
the top, but two levels high. For example:
In a discussion of these volumes, hopefully your students will notice that these each have twice
as many blocks as they would if they were only one level high. Encourage them to calculate the
volume of the first as 2 × 3 × 2 = 12 cubes, the area of the second as 3 × 3 × 2 = 18 cubes, etc.
The next step is to encourage the class to build structures like these, but with any number
of levels. This is where we are likely to run out of blocks, so it will probably need to be done by
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only 2 or 3 groups of children. If you are very limited with blocks, perhaps you will have to
make one big structure for all the class to see, for example:
Hopefully, your students will be able to work
out that the volume of the overall structure is 3 × 4 ×
7 = 84 blocks. Ideally, they will go straight to
multiplication, but they might begin by calculating
that each layer uses 3 × 7 = 21 blocks and then
multiply this by 4. If they got really good at the twolayer stacks, they might notice that this is two of
them on top of one another.
In any case, the next level of abstraction is to introduce some boxes on paper where the
lengths are written out, but the individual blocks are not illustrated. For example:
The volume of this block can be found by multiplying
10 × 8 × 5 to get 400 cubes. If we include the units, then we get
10 in u 8 in u 5 in = 400 in u in u in = 400 in 3 . A block that
measures 1 inch on each side is called a cubic inch, thus this
box has a volume of 400 cubic inches.
It might take a little longer to run through these exercises than to merely tell the class that
the volume formula for a box is: Volume = length × width × height. However, it will benefit
your class greatly for them to see the origins of this formula on their own.
It will help to explain to the class that an arrangement with one layer will have the same
number for its bottom area as it does for its volume. This is because each square of area on its
base will correspond with a cube of volume:
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Rather than look at the volume of a box as Volume = length × width × height, we could
instead look at it as Volume = Area of base × height. In fact, because the base of a box is a
rectangle, the Area of base = length × width anyway. For example, returning to an earlier
example:
The base of this box is a rectangle that measures 3 by 7, thus has an area of 21 squares.
The box has a height of 4, and thus the volume is 21 × 4 = 84 cubes.
The advantage of this method is that it works for all prisms. A prism is a solid object that
starts with a flat base and is made of identical layers like this. For example:
The first object above has a top and a bottom that are both rectangles. The technical term for this
shape is a rectangular prism, but most people call them boxes. The second object above has
identical triangles for its top and bottom. This is called a triangular prism. The last shape has a
circle for its top and bottom (they don't look like circles only because they are drawn as if
viewed from the side). This ought to be called a circular prism, but instead is called a cylinder.
With some dimensions, we can figure out their volumes:
The formula for all three is: Volume = Area of base × height.
For the box, the area of the base is 3 cm u 8 cm 24 cm 2 and the height is 5 cm, so the
volume is 24 cm 2 u 5 cm 120 cm 3 .
For the triangular prism, the base is a triangle with a height of 3 in and a base of 14 in
(you might have to turn the paper to see this), thus the area of the base is 12 u 3 in u 14 in 21in 2 .
Because the prism is 4 inches tall, we can imagine that it is made of 4 layers, each with 21 cubic
inches of volume, for a total volume of 21in 2 u 4 in 84 in 3 .
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The base of the cylinder is a circle with radius 6 ft, thus the area is Area of a circle = ʌ ×
radius × radius = ʌ × 6 ft × 6 ft = 36 × ʌ ft2 § 36 × 3.14 ft2 = 113.04 ft2. The volume of the
cylinder is thus Volume of prism = Area of base × height § 113.04 ft 2 u 10 ft 1,130.4 ft 3 .
Many books use the formula: Volume of cylinder = ʌ × radius × radius × height.
However, it is better to teach your children the more general formula for the volume of a prism
instead (or in addition). This requires less memorization, can be used in lots of different
situations, and makes some sense, especially if you start with building blocks of various layers.
The next sort of volumes we can calculate are those which start from a flat base and then
taper up to a point. Mathematicians call all of these cones, but most people use that term only
for the ones with circular bases. When the base is a square or a rectangle, they are often called
pyramids:
To explore shapes like these, we will compare them to prisms of the same height and
base. The easiest way to do this is to begin comparing cones and cylinders. If you start with an
empty 15-ounce can (a standard size for canned vegetables), it will be about 11 cm tall and 7.5
cm in diameter. In order to make a cone with the same dimensions, start by drawing a line 11.6
cm along the bottom of a piece of paper (oak tag would be even better). Next, use a protractor to
measure a 116° angle from one end of this line. Finally, use a protractor, ruler-compass, or piece
of string to draw a sector with this angle and radius. It wouldn't hurt, also, to add a little flap
along one edge:
Cut this shape out. Fold along the line
that separates the sector from the flap. Roll up
the sector into a cone, lining the straight edge
up with the fold. Tape the fold down onto the
outside of the cone. It is all right to lightly fold
the cone flat in order to line up the two edges
precisely – you can always squeeze it into a
more proper cone shape later.
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This requires some precision (and
maybe even some practice and failure), so you
might want to make up a number of cones in
advance, rather than leave the task to your
students.
For the activity, break the class up into groups. Each group gets one cone, one empty
can, a spoon, a plastic tub, and some sort of small, dry material, for example uncooked rice or
oatmeal. One member of the group will hold the cone (trying to make it as rounded as possible),
while another member slowly spoons the oatmeal (or whatever) into the cone. If they hold
everything over the tub, then clean-up ought to be easy. When the cone is completely full (level
across the top and not heaping over), the students should carefully pour all of its contents into the
can. The group should then repeat this process as many times as possible (perhaps taking turns
with the roles of holding the cone, spooning the oatmeal, and pouring into the can).
If everything is done correctly (the cones are the right size and filled correctly, nothing is
spilled, etc.) then three cones of material will exactly fill the can (if the group is using oatmeal,
they might need to pack it down a little). Two cones will certainly fall short and four cones will
definitely overflow the can:
We can conclude that the volume of a cone is one-third that of a cylinder with the same
height and diameter. Thus Volume of a cone = 13 u ʌ × radius × radius × height.
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In fact, anything that tapers to a point like this will be exactly one-third the volume of a
prism with the same height and base-area. For example, Volume of a cone = 13 × area of
circular base × height. Similarly, the volume of a pyramid is Volume of pyramid =
1
3
× area of
base × height.
For example, using the following objects:
The area of the base of the cone is ʌ × 2 cm × 2 cm § 12.56 cm2 and the height is 6 cm,
thus the volume is approximately 13 u 12.56 cm 2 u 6 cm 25.12 cm3 .
The square-bottomed pyramid has a base area of 3 in × 3 in = 9 in2 and a height of 4 in,
thus it has a volume of exactly 13 u 9 in 2 u 4 in 12 in 3 .
The rectangular-bottomed pyramid on the right has a base area of 4 ft × 12 ft = 48 ft2 and
a height of 5 ft, so it has a volume of 13 u 48 ft 2 u 5 ft 60 ft 3 .
Most books cover these formula separately. However, the formula Volume of a cone or
pyramid = 13 × area of base × height is easier to remember and use.
The final volume formula which is often taught to elementary school children is the
volume of a sphere. This is a bit more difficult to illustrate, mostly because it is not easy to make
or find a sphere which can be filled with material. If you could find a coconut that looked very
round, you could cut it in half with a hacksaw and an hour or so of hard labor. You could then
scoop out the coconut meat, clean it up, and use that. Unfortunately, coconuts are not precisely
spherical. However, your students might have so much fun playing with the shell that they'd
forgive whatever slight errors might occur in the process. For a little bit more money than a
coconut (but less than $20), plastic hemispheres can be purchased online.
In any case, to make the demonstration you will need to have a hemisphere, but two
would be even better. Measure the diameter of the inside of the sphere (this is important,
especially if your sphere has a significant thickness, as will be the case with a coconut. Divide
this number by 2 to get the radius, and then multiply by 1.414. Draw a circle with this radius on
a piece of oak tag. Next, use a protractor to measure a sector of 255° out of this larger circle. If
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you cut out this and tape it together into a cone, it should have the same height and radius as your
hemisphere:
As soon as you have a hemisphere and a cone with the same height and radius, you can
conduct the demonstration. Just as with the cone and cylinder, you fill the cone with dry oatmeal
or rice, and then pour it into the hemisphere. This time, if everything goes well, two cones will
fill the hemisphere exactly:
This means that four cones will fill a full sphere:
The volume of each cone can be calculated using the radius and the height (which is the
same as the radius). Each cone has volume 13 × ʌ × radius × radius × radius = 13 × ʌ × radius3.
The volume of the sphere is four times bigger.
4
4
We conclude: Volume of a sphere =
× ʌ × radius × radius × radius = × ʌ × radius3.
3
3
Questions:
(1) Build a cone with the same height and volume as a standard 15-ounce can. Use it to
demonstrate the formula for the volume of a cone.
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