WEBVTT

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What we're going to
explore in this video

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are polyhedra, which is just
the plural of a polyhedron.

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And a polyhedron is a
three-dimensional shape

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that has flat surfaces
and straight edges.

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So, for example, a
cube is a polyhedron.

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All the surfaces are flat, and
all of the edges are straight.

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So this right over
here is a polyhedron.

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Once again, polyhedra is plural.

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Polyhedron is when
you have one of them.

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This is a polyhedron.

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A rectangular pyramid
is a polyhedron.

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So let me draw that.

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I'll make this one a little
bit more transparent.

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Let me do this in a
different color just for fun.

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I'll make it a magenta
rectangular pyramid.

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So once again, here I
have one flat surface.

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And then I'm going to have
four triangular flat surfaces.

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So this right over here, this
is a rectangular pyramid.

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Now, it clearly
looks like a pyramid.

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Why is it called a
rectangular pyramid?

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Because the base right
over here is a rectangle.

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So these are just a few
examples of polyhedra.

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Now, what I want to think
about are nets of polyhedra.

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And actually, let me draw
and make this transparent,

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too, so we get full appreciation
of the entire polyhedron,

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this entire cube.

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So now let's think about
nets of polyhedron.

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So what is a net
of a polyhedron?

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Well, one way to think about it
is if you kind of viewed this

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as made up of cardboard, and you
were to unfold it in some way

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so it would become
flat, or another way

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of thinking about
it is if you were

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to cut out some
cardboard or some paper,

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and you wanted to fold it up
into one of these figures,

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how would you go about doing it?

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And each of these polyhedra
has multiple different nets

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that you could create so
that it can be folded up

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into this
three-dimensional figure.

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So let's take an example.

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And maybe the simplest example
would be a cube like this.

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And I'm going to color code it.

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So let's say that the bottom of
this cube was this green color.

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And so I can represent
it like this.

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That's the bottom of the cube.

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It's that green color.

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Now, let's say that this back
surface of the cube is orange.

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Well, I could
represent it like this.

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And notice, I've kind
of folded it out.

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I'm folding it out.

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And so if I were to flatten it
out, it would look like this.

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It would look like that.

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Now, this other backside,
I'll shade it in yellow.

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This other backside right over
here, I could fold it backwards

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and keep it connected along
this edge, fold it backwards.

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It would look like this.

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It would look like that.

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I think you get the
general idea here.

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And just to be clear,
this edge right over here

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is this edge right over there.

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Now I have to worry
about this top part.

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Maybe it is in-- let me
do it in a pink color.

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This top part of the cube
is in this pink color,

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and it needs to be attached
to one of these sides.

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I could attach it to
this side or this side.

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Let's attach it over here.

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So let's say it's attached to
that yellow side back here.

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So then when we
fold it out, when

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we really unpack the thing,
so we folded that yellow part

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back, then we're
folding this part back,

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then it would be
right over here.

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And then we could fold this
front face right over here.

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We could fold that
out along this edge,

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and it would go
right over there.

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It would go right over there.

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And then we have one
face of the cube left.

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We have this side
right over here.

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Well, we could do,
actually, several things.

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We could fold it
out along this edge.

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And then we would draw the
surface right over there.

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Or if we want to do
something interesting,

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we could fold it
out along the edge

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that it shares with the
yellow, that backside.

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So we could fold
it out like this.

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So if we folded
it out like this,

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it would be connected to the
yellow square right over here.

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So you see that
there's many, many ways

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to construct a net
or a net that when

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you fold it all back up will
turn into this polyhedron,

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in this case, a cube.

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Let's do one more example.

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Let's do the rectangular
pyramid, because all of these

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had rectangles.

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Or in particular, these had
squares as our surfaces.

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Now, the most
obvious one might be

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to start with your
base right over here.

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Start with your base and
then take the different sides

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and then just fold
them straight out.

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So, for example, we could take
this side right over here,

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fold it out, and it
would look like that.

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We could take this
side back here,

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and once again,
just fold it out.

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And it would look like that.

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It should be the same
size as that orange side,

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but I'm hand drawing it, so
it's not going to be perfect.

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So that's that right over there.

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And then you could take this
front side right over here,

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and once again, fold
it out along this edge.

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So it would look like this.

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And then finally, you could
take this side right over

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here, and once again, fold
it out along this edge

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and it would go right there.

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But this isn't the only net
for this rectangular pyramid.

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There's other options.

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For example, and just
to explore one of them,

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instead of folding that
green side out that way,

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instead we might have
wanted to fold it out

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along this edge with the
yellow side that you can't see.

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Actually, let's make it
a little bit different.

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Let's fold it out along this
side since we can see the edge.

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And let me color the edge.

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So this is the edge right over
here on the blue triangle.

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So this is the edge.

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And when you fold the
green triangle out,

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it would look like this.

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If you fold it the
green triangle out,

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it would look like this.

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So hopefully this gives
you an appreciation.

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There's multiple ways to
unfold these three-dimensional

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figures, these polyhedra,
or multiple ways

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if you wanted to do
a cardboard cutout

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and then fold things back
together to construct them.

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And these flattened
versions of them,

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these things, these unpacking of
these polyhedra, we call nets.