♪
- [female voice]Wow! That's a lot of cardboard!
Hi! It's Charday, and this morning I'm at Hawkeye
Corrugated Box Company. They produce and design boxes for
every kind of product imaginable. From these funny looking little boxes
to huge boxes big enough to fit a car! I bet you
use boxes every day, but never stop to think about how they're made.
It's all about math. In fact, the construction of boxes
relates to a topic in geometry, called nets. They're super
cool! I'll show you! And then, we can box up another
problem solved. Here at Hawkeye Box, they
design the boxes, print the boxes, construct the boxes and
they ship the boxes! When they're working on a new box for a
client, they have to consider several things. First, what the box
is used for will determine its shape and dimensions. If it's too big,
it will waste material and will cost too much. If it's too small, the
product won't fit inside. Sometimes, there are other considerations,
such as how the boxes will be displayed and stacked,
the size of each face for advertising, and the function of the box.
Cereal boxes like this, fit into
your hand. You wouldn't want to pour your cereal out of this.
See what I mean? Regardless of their size or shape, each box
starts out as a pattern or a net. Nets are two
dimensional figures that can be folded into three-dimensional shapes.
We can use this box, which is a cube, to make sense
of nets. I have traced each face of this box
onto cardboard to show you what the net or pattern looks like.
See? Here at Hawkeye box, they use computer programs to design
the nets for their boxes. A cube
has six square faces. So,
the net is made of six squares.
♪
Each square shares a side
with at least one other square.
Hawkeye has to add tabs to keep the box together.
But you get the idea! Here's a pattern for the cube I
just folded. See how it comes together?
Here is another net for a cube.
Try to visualize how it will fold.
♪
Check out this one! Try to visualize how it will fold.
♪
However, not every
arrangement of six squares can be folded into a cube.
Try to visualize what happens when you fold this. It doesn't
work! That's a problem.
Most boxes are not cubes, they're rectangular
prisms. Let's consider four important factors when working
with rectangular prisms. The net must have the same number
of rectangles as faces on the prism. For example,
this prism has six rectangular faces.
One, two, three, four, five,
and six. Got that? Now look
at this pattern. Is it a net for a rectangular prism?
No. There are only five faces, so it won't
work. When you fold it, one end is missing.
The net for rectangular prisms must have three
pairs of congruent rectangles. The top and the bottom are
congruent or exactly the same size and shape. The
front and the back are congruent and the two ends are congruent.
Here they are on the net.
How about this pattern? It only has
two pairs of congruent rectangles.
These two rectangles are not the same size and shape.
When you try to fold it, you get this.
It doesn't quite work. Here's
another thing to remember about rectangular prisms. When they're folded,
the congruent rectangles are opposite faces.
Remember, the top and the bottom, the front and the
back and the two ends.
This looks like a net for a rectangular prism. It has
six faces and three pairs of congruent faces, but
won't work because one of the two congruent rectangles is not
in the correct position. This is what you get;
it's not a net either.
You also need to pay attention to the edges.
Let me show you another pattern! There are three
pairs of congruent rectangles and it looks like the congruent
rectangles will be opposite each other when folded. But can you tell
what's wrong? Look at these edges.
When the pattern is folded, these edges just don't
match. That's the fourth key point for working with
nets for rectangular prisms. When folded,
connecting edges must be the same length. There are
many different nets that could work for rectangular prisms, but
sometimes the nets can be a little complicated. It's important
to be able to visualize how they will
fold. This is a net for a rectangular prism. Will it still fold
into a rectangular prism if we move these two faces?
Try to visualize it.
It does! If you move
one face back to the original position, the net still folds correctly!
Visualize
what would happen if we move and rotate one face.
♪
The net still folds into a rectangular
prism. Today, we've looked at cubes and
rectangular prisms. But, any three dimensional shape
or box has a net. It's important to be able to visualize
folding nets to help you put a lid on geometry.
Problem solved. I wonder what the nets
for these look like!