5
CHAPTER 5
THE TRIANGLE
BESTIARY
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In medieval times, people wrote books called “bestiaries” about weird and
wonderful beasts and what lessons one could learn from them. Triangles
are deceptively simple and show up where you least expect them. Every
shape that you can make with straight sides can be broken into triangles.
They make up the surface of a 3D print, the structure of a bridge, and the
roof of a house. Let’s look for these beasts in the wild, see what makes them
operate the way they do, and analyze their habitat.
WHAT IS A TRIANGLE?
A triangle (Figure 5-1) is the simplest way you can make a shape with straight
lines that has a distinct inside and outside. It is made of three lines. (Usually
those lines are straight, but we saw an example of when they aren’t with the
Reuleaux triangle in Chapter 4.) Mathematicians call the points where the
lines meet the vertices (just one is called a vertex).
ANGLES OF A TRIANGLE
Back in Chapter 3, we learned about angles and how they are defined. Tri-
angles have a special relationship with some angles. Triangles with one 90°
angle are called right triangles, and a little later in this chapter we will learn
FIGURE 51: The anatomy of a triangle
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3D Printable Models Used in this Chapter
See Chapter 2 for directions on where and how to download these
models.
TriangleAngles.scad
Demonstrates that the angles of a triangle sum to 180 degrees
ExtrudedTriangle.scad
Just two lines long, this model draws a triangle from any three points
and extrudes it to the desired thickness.
TriangleSolver.scad
Creates three similar triangles (scaled). They can be specified using
one of several rules, described in the text.
TriangleArea.scad (TWO SETS needed)
This model consists of a triangle broken into several pieces. It has an
option to print out one or two triangle sets. Set the variable to TRUE to
make two sets, or FALSE to make one.
TriangleBox.scad
This model creates a rectangular open box to support the TriangleArea
triangles.
TriangleAreaBox.scad
This model creates a triangular open box that holds together just one
TriangleArea triangle.
Other supplies for this chapter
• Cardboard
• Paper clips
• Protractor
• 0.2mm stretch cord
• Drinking straws
• A few pieces of construction
or similar heavy paper
• 3 rulers (we used 8-inch
ones, but length isn’t
important)
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more about special ratios of their angles and sides.
Two times 90° is 180°, and a straight line is a 180°
angle. If you kept going all the way around to bring
your hand back to where you started, you would
have turned 360°, or a full circle.
If you think about it, as one angle of a triangle gets
narrower, the triangle will be pinched together
at that vertex, but the other two angles will need
to get correspondingly bigger so that it stays a
triangle. We can either use the 3D printed model
TriangleAngles.scad or just a triangle cut out of
paper to demonstrate that, as it turns out, the angles inside a triangle always add up to 180 degrees.
Let’s use a very old trick that dates back to Euclid (who lived around 2400 years ago in Greece).
PROVE TRIANGLE ANGLES ADD TO 180 DEGREES
Make a triangle cut into four pieces, either by 3D printing the TriangleAngles.scad model or using paper.
To make a paper version, first cut out any triangle you like. Then, using a compass, cut off each of the
triangle’s vertices (as we show with the 3D printed triangle in (Figure 5-2)). You’ll need to keep track of
which corner of each piece was a vertex of the original triangle, and making a curved cut as we have
done here will make that easier.
Now remove these three vertex pieces from the triangle and put the three vertices together, so they
make a straight line (and thus add up to 180 degrees). You can see this in Figure 5-3.
The longest side will have the biggest angle directly across from it, and the smallest side will have the
smallest angle across from it. Of course, that means that if two angles are equal, the sides opposite
FIGURE 52: The triangle assembled
FIGURE 53: Figure 5-3: The triangle’s vertices rearranged to
show that the angles sum up to a straight line.
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them will be the same length. Print, or cut out,
several variations of this model (changing the
angles each time) to get some intuition about
why this is true. You’ll see that if some of the
angles get wider, the others have to become
correspondingly smaller so that the sum of the
angles will remain the same.
For right triangles (with, by definition, one 90°
angle) this means that the other two angles
have to add up to 180° - 90°, or 90°. Since each
of these other angles has to be less than 90°,
the side that does not touch the 90° angle will
be the longest. This longest side of a right tri-
angle (the one opposite the right angle) is called
the hypotenuse. The longest side (opposite the
biggest angles) doesn’t have a special name in
other triangles.
SPECIFYING TRIANGLES
How many angles and/or sides of a triangle do
you need to know to pin down all its dimensions
and angles? Try to figure it out before you read
the answer in the next section.
CONGRUENT
TRIANGLES
Congruent triangles are the same size and
have the same angles as each other (although
they can be oriented differently). If we 3D print
a triangle or cut one out of a piece of paper,
it is pretty obvious that if we take this plastic
or paper triangle and flip it over (mirror it),
rotate it around its center, or move it (which
mathematicians would call “translating”), the
triangle does not change. If we were to do any
of those things and trace around the triangle in
the starting position and then trace around the
triangle after we had rotated, translated, and/
or mirrored it, each of those traced triangles
would be congruent to the one we started with.
How to use a protractor
A protractor is a device for measuring angles. You
can print one out (search online for “download pro-
tractor”) or you can buy one. If you buy one, a clear
plastic one is handy because it is easier to see what
you are doing.
If you are measuring the angles of a triangle with a
protractor, first put the vertex of the angle you are
measuring on the crosshairs at the bottom of the
protractor. Line up the bottom of the angle with the
line on the bottom of the protractor(Figure 5-4).
Then you can read off the angle from the scale
around the edge. Either read up from the bottom
if the angle is less than 90° (as in this case, where
the angle is about 47°) or the outer scale if it is more
than 90°. Note that how carefully we can measure
comes down to how good our tools are. The width
of the lines making up your triangle, how accurately
your protractor is printed, and how good you are at
estimating will all come into play. A plastic protrac-
tor like the one shown is probably good to plus or
minus half a degree.
FIGURE 54: Measuring an angle with a protractor. (Base
of the angle is parallel to the line across the bottom of the
protractor.)
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