Let’s try an experiment to see if it is true that the ratio of the volumes of a cone,
half-sphere, and cylinder is 1 : 2 : 3 (when the radius of the cone and cylinder are
equal to their height). First, fill the cone with something that won’t pack down,
like granulated sugar or salt. Fine sand will work, too. Powders like flour or pow-
dered sugar pack down, so they will be less precise for this purpose.
Of course, water would be the obvious choice, but it’s sometimes hard to
get 3D prints to hold water around seams, depending on how well-tuned the
printer is. The surface tension of water also makes it more difficult to fill to a
precise level, and to avoid spilling. For these reasons, we suggest something
granular instead. We show the process in Figures 8-9 through 8-14 using
Himalayan pink salt. (Ignore the fact that we spilled a little - you will, too, and
the differences will be smaller than can be measured.)
FIGURE 87: The cone, sphere, and cylinder molds (back row) and corresponding positive
models (front row).
FIGURE 88: Positive models fitted into their respective molds
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If the radius and the height of the models are not equal, the relationships get
more complicated. We will see in the next section on pyramids that the rela-
tionship of 1:3 in volume between a cone and a cylinder holds as long as the
cone and cylinder have the same base area and height as each other.
Once you finish this experiment, save the granular material you were mea-
suring so we can find its density later in this chapter.
VOLUME OF PYRAMIDS
We would like to figure out the volume of a pyramid, which is a shape (like
the Egyptian pyramids) that is similar to a cone, but with flat triangular sides
(Figure 8-15). The base is a polygon.
Let’s imagine that a pyramid and a cone have the same base area and the
same height as each other, as is the case for the two in Figure 8-15. Can we
relate their volume? Cavalieri figured out that, in fact, they are equal.
Imagine you have two planes that are parallel to each other. Let’s say that
FIGURE 89: Fill the cone mold. FIGURE 811: Add the contents of the
cone to what is already in the half-
sphere. The half-sphere should just
exactly fill.
FIGURE 810: Pour the contents of the
cone mold into the half-sphere, then fill
the cone again.
FIGURE 812: Now, get your cylinder.
Pour the contents of the half-sphere
into it.
FIGURE 813: Then fill the cone a third
time.
FIGURE 814: Pour the contents of
the cone and the half-sphere into the
cylinder (1 + 2 = 3). You will see that the
cone + half-sphere fills the cylinder.
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one plane includes the base of the pyramid and cone, and the second plane
is just where the cone and pyramid come to a point (that is, their cross-sec-
tional areas go to zero).
We already said that the base areas of the two are the same, and they are the
same height, so their cross-sectional area goes to zero at the same height as
each other. We can also argue that since the cross-sections start the same
and are continuously decreasing as you go up at the same rate, they will be
equal everywhere. Thus, by Cavalieri’s principle, they are equal. So we know
now that the volume of a pyramid is ⁄ base area * height. Note that it doesn’t
matter how many sides the pyramid has.
FIGURE 815: A pyramid (left) and cone (right)
FIGURE 816: Pyramid and cone with the same volume
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There’s another way to think about it, too. In Chapter 7, we saw that as the number of sides of a cir-
cumscribed or inscribed polygon goes up, it approaches being its circle. So we can think of a cone as a
pyramid with a very large number of sides. As long as we are holding the cross-sectional area constant,
the volume will stay constant as we increase the number of sides. In fact, 3D models like the ones we
used above can’t have perfectly smooth curves, because of how the file format works. What look like
circles are actually regular polygons with straight sides, the sides are just so small that you can’t see
them (as we saw in Chapter 3).
Try creating a pyramid and cone with the same volume, as we saw in Figure 8-16. Fill one of them to just
level with salt or sand (Figure 8-16). Now pour the material into the other one. The same, aren’t they?
These were created with pyramid.scad volume (v) of 25,000 cubic mm and a height (h) of 50 mm.
COMPARING VOLUMES
Now that you you can calculate the volume of cones, pyramids, and prisms, try printing out several that
are the same volume to help your intuition. Here’s a summary of what we have learned so far in this
chapter.
Shape Volume
Sphere ⁄πr
3
Cylinder πr
3
h (or base * height)
Cone ⁄πr
2
h (or ⁄base * height)
Prism base * height
Pyramid ⁄base * height
Try printing a prism that is the same volume as a cone, but 1/3 the height. The base area should then be
the same. A prism with the same base and height as a cone will have 3 times the volume.
IF YOU DON’T HAVE A 3D PRINTER
If a 3D printer isn’t available, you can make these parts out of things you have lying around, although it
will be challenging to get them precisely in the ratio here.
Cylinders are easy to find (straight-sided cups or glasses, or soup cans will work). Cones are easy to
make out of paper and should self-straighten into a better cone when filled, since the correct shape is
the maximum volume for its surface.
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A hemisphere is a little tricker. Ping pong balls cut in half would be good, or
the bottom of a plastic easter egg is probably close enough. Craft stores also
have holiday ornament balls that split in two.
Of course, in any of these cases, accurately measuring the internal dimen-
sions will be the hard part, and you won’t get them to all be equal radius and
height like the printed ones. Also, for most of the hemisphere options, you
will need to measure the base radius and height separately, since they may
not be equal (which means results will be a bit off). In this case, you’ll have to
use ⅔ of the base area times the height for the volume of your hemispheroid.
CAVALIERI’S PUZZLE
Earlier in this chapter, we said that the volume of a cone or a pyramid is 1/3
the base times the height. To try to convince you a bit more than we can with
moving sand around between models, let’s try out a puzzle. We can show
an example of a cube that is created from three identical pyramids, with the
model 3_pyramid_puzzle.scad. Each of the pyramids has a base equal to
a side of the cube, and so collectively they should add up to the cube (Figure
8-17).
The only parameter in 3_pyramid_puzzle.scad is the length of a side of the
cube, the variable size, in mm. Since relative dimensions are crucial, if you
want to scale this model, you can do so by changing size, or by scaling it
(uniformly) in your 3D printer’s slicing program. If you scale one axis more
than another, the pieces won’t fit.
Try rearranging these pyramids into a cube to prove that it works. As a hint,
note that the square sides need to be on the outside, and some sides of the
FIGURE 817: The three pyramids that together have the same volume as a cube
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