Nets of polyhedra

What we're going to
explore in this video

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

And a polyhedron is a
three-dimensional shape

that has flat surfaces
and straight edges.

So, for example, a
cube is a polyhedron.

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

So this right over
here is a polyhedron.

Once again, polyhedra is plural.

Polyhedron is when
you have one of them.

This is a polyhedron.

A rectangular pyramid
is a polyhedron.

So let me draw that.

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

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

I'll make it a magenta
rectangular pyramid.

So once again, here I
have one flat surface.

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

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

Now, it clearly
looks like a pyramid.

Why is it called a
rectangular pyramid?

Because the base right
over here is a rectangle.

So these are just a few
examples of polyhedra.

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

And actually, let me draw
and make this transparent,

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

this entire cube.

So now let's think about
nets of polyhedron.

So what is a net
of a polyhedron?

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

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

so it would become
flat, or another way

of thinking about
it is if you were

to cut out some
cardboard or some paper,

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

how would you go about doing it?

And each of these polyhedra
has multiple different nets

that you could create so
that it can be folded up

into this
three-dimensional figure.

So let's take an example.

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

And I'm going to color code it.

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

And so I can represent
it like this.

That's the bottom of the cube.

It's that green color.

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

Well, I could
represent it like this.

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

I'm folding it out.

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

It would look like that.

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

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

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

It would look like this.

It would look like that.

I think you get the
general idea here.

And just to be clear,
this edge right over here

is this edge right over there.

Now I have to worry
about this top part.

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

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

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

I could attach it to
this side or this side.

Let's attach it over here.

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

So then when we
fold it out, when

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

back, then we're
folding this part back,

then it would be
right over here.

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

We could fold that
out along this edge,

and it would go
right over there.

It would go right over there.

And then we have one
face of the cube left.

We have this side
right over here.

Well, we could do,
actually, several things.

We could fold it
out along this edge.

And then we would draw the
surface right over there.

Or if we want to do
something interesting,

we could fold it
out along the edge

that it shares with the
yellow, that backside.

So we could fold
it out like this.

So if we folded
it out like this,

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

So you see that
there's many, many ways

to construct a net
or a net that when

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

in this case, a cube.

Let's do one more example.

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

had rectangles.

Or in particular, these had
squares as our surfaces.

Now, the most
obvious one might be

to start with your
base right over here.

Start with your base and
then take the different sides

and then just fold
them straight out.

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

fold it out, and it
would look like that.

We could take this
side back here,

and once again,
just fold it out.

And it would look like that.

It should be the same
size as that orange side,

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

So that's that right over there.

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

and once again, fold
it out along this edge.

So it would look like this.

And then finally, you could
take this side right over

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

and it would go right there.

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

There's other options.

For example, and just
to explore one of them,

instead of folding that
green side out that way,

instead we might have
wanted to fold it out

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

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

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

And let me color the edge.

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

So this is the edge.

And when you fold the
green triangle out,

it would look like this.

If you fold it the
green triangle out,

it would look like this.

So hopefully this gives
you an appreciation.

There's multiple ways to
unfold these three-dimensional

figures, these polyhedra,
or multiple ways

if you wanted to do
a cardboard cutout

and then fold things back
together to construct them.

And these flattened
versions of them,

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