prove that the three angles are the same in all three triangles is to overlay
them, with the relevant angle lined up.
Note that similar triangles have the same set of angles, but are not (neces-
sarily) congruent because they may be different sizes, as we see here. If the
sizes of the triangles were the same, they would indeed be congruent, so
congruent triangles are a special case of similar triangles. Here we can see
these three triangles all have at least one angle in common (Figure 5-16).
Mixing the three scaled triangles still allows you to create a straight line out of
the angles, just as we saw in the exercise where we lined up the three angles of
one triangle earlier in the chapter. In other words, the angles add up to 180° even
if you use the angles from the same triangle scaled up or down.
To test this out for yourself, take a set of similar triangles created by Trian-
g l e S o lv e r.s c a d (or that you create with paper and a protractor) and show
that even though the scales of the triangles are different, all three angles are
the same (Figure 5-17). We took three similar triangles (which each have the
same three angles as each other), and we turned each triangle so that we
FIGURE 516: Similar triangles lined up to show their angles are the same.
FIGURE 517: The three similar triangles in Figure 5-16 rearranged to show that taking one
angle from each triangle still adds up to 180°.
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used one different angle from each one. So, it is equivalent to cutting off the
three angles of one triangle and aligning them, as we did in Figure 5-2 and 5-3.
AREA OF A TRIANGLE
We’ve been talking a lot about a right triangle. Try folding a piece of square
paper along its diagonal to make a right triangle (Figure 5-18).
The area of the square piece of paper is just the lengths of the bottom (or top)
and a side multiplied together. Since you folded one half of the paper over on
itself, the area of this triangle is just one-half of the area of the square piece
of paper you started with.
But what about all the other triangles? Let’s demonstrate that the same general
procedure applies more broadly no matter what the shape of the triangle.
TRIANGLE AREA MODELS
We have created several models that you can use together to visualize how
several different triangles, all of which have the same base and height, will
occupy half of a rectangle with the dimensions base * height. We do that by
slicing up the triangles so that it is possible to arrange two of them into the
rectangle. If two of them fit exactly into the rectangle, it follows that each
of them is half of the area of the rectangle. For this to work, some triangles
need to be cut into several pieces, as we will see. You will need to print the
following (or make paper equivalents):
FIGURE 518: Making a right triangle from a square piece of paper
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• TriangleArea.scad (TWO SETS needed)
• This model consists of a one- or multi-part triangle.
It has an option to print out one or two triangle sets.
Set the variable twosets to TRUE to make two sets
at once, or FALSE to make one (if, for example, you
want to print them in different colors, as we have
done here).
• TriangleBox.scad
• This model creates a rectangular open box of the
same base and height as the TriangleArea triangles.
• TriangleAreaBox.scad
• This model creates a triangular open box that holds
together just one TriangleArea triangle
These models all have some combination of the following variables:
• base = 105
• the base of the triangle, in mm
• height = 70
• the height of the triangle, in mm
• thickness = 20
• the thickness of the extruded triangle models, in
mm
• top = base * 1.3
• the position of the highest point of the triangle above
the base
• linewidth = 0.1
• If you set thickness = 0, the program will create a
file you can export as a .svg file to print on paper.
This variable is the width of lines drawn, in mm.
If you have to change the sizes of these triangles, it is better to change these
numbers in OpenSCAD rather than to scale the models in a slicer because
tolerances for parts to fit together may not work right if you scale every-
thing up or down uniformly. Note that these models are designed to all work
together and in some cases be fitted together. If you change one, you need to
change them all.
If you change the position of the highest point (the value of the parameter
top) you may get more or fewer pieces per triangle. (See the section “Paper
Version of Triangle Area Model” to see the parameter values that will give
you a version you can print on paper.)
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We will walk through three examples of varying complexity to give you some
ideas on how to use the model, from a very simple one through a bit of a
puzzle.
RIGHT TRIANGLE MODEL
First, let’s test whether two right triangles, similar to our folded square
piece of paper, will indeed make a rectangle. First, set the parameter top in
TriangleArea.scad and TriangleAreaBox.scad to:
top = base * 1.0;
This will create a right triangle, with its top vertex directly over the right-hand
side of the model, as we see in Figure 5-19.
Then 3D print these models, leaving the other parameters at their defaults
noted earlier:
Two sets of the triangle from TriangleArea.scad (or you can do this
by setting the parameter twosets to TRUE) to do both at once).
One triangular box with TriangleAreaBox.scad (optional; we’ll see
why shortly)
One of the rectangular boxes from TriangleBox.scad
This triangle prints in just one piece. In Figure 5-19 we see the triangle from
TriangleArea.scad in the organizing box made by TriangleAreaBox.scad.
The box isn’t really useful here, but we show it here for consistency with the
more complex cases we will do in the next section.
FIGURE 519: Right triangle in its box
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Next, we created an open-top rectangular box with the same base and height
as our triangle, using TriangleAreaBox.scad. In Figure 5-20 we see one green
triangle model in the blue box. In Figure 5-21, we see that if we add a second,
identical triangle, they exactly fill the box. (The boxes are slightly bigger than
the triangle pieces so they are easy to take in and out.)
ACUTE TRIANGLE MODEL
Now, let’s see if we can do the same thing with an acute triangle. Set the
parameter top in TriangleArea.scad and TriangleAreaBox.scad to:
FIGURE 520: One triangle in a box with the same base and height
FIGURE 521: Two triangles fitting in the same box
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