Volume of a rectangular prism with fractional cubes

So I have this
rectangular prism here.

It's kind of the shape of
a brick or a fish tank,

and it's made up of
these unit cubes.

And each of these unit
cubes we're saying

is 1/4 of a foot by 1/4 of
a foot by 1/4 of a foot.

So you could almost
imagine that this

is-- so let me write it
this way-- a 1/4 of a foot

by 1/4 of a foot
by 1/4 of a foot.

Those are its length,
height, and width,

or depth, whatever
you want to call it.

So given that,
what is the volume

of this entire rectangular
prism going to be?

So I'm assuming you've
given a go at it.

So there's a couple of
ways to think about it.

You could first think about
the volume of each unit cube,

and then think about how
many units cubes there are.

So let's do that.

The unit cube, its
volume is going

to be 1/4 of a foot times 1/4
of a foot times 1/4 of a foot.

Or another way to think about
it is it's going to be 1/4 times

1/4 times 1/4 cubic
feet, which is often

written as feet to the
third power, cubic feet.

So 1/4 times 1/4 is
1/16, times 1/4 is 1/64.

So this is going to be 1
over 64 cubic feet, or 1/64

of a cubic foot.

That's the volume
of each of these.

That's the volume of
each of these unit cubes.

Now, how many of them are there?

Well, you could view them
as kind of these two layers.

The first layer has 1,
2, 3, 4, 5, 6, 7, 8.

That's this first
layer right over here.

And then we have the
second layer down here,

which would be another 8.

So it's going to
be 8 plus 8, or 16.

So the total volume
here is going

to be 16 times 1/64
of a cubic foot, which

is going to be equal to
16/64 cubic feet, which

is the same thing.

16/64 is the same thing as 1/4.

Divide the numerator and
the denominator by 16.

This is the same thing
as 1/4 of a cubic foot.

And that's our volume.

Now, there's other ways that
you could have done this.

You could have just thought
about the dimensions

of the length, the
width, and the height.

The width right over here
is going to be 2 times

1/4 feet, which is
equal to 1/2 of a foot.

The height here
is the same thing.

So it's going to be 2
times 1/4 of a foot, which

is equal to 2/4,
or 1/2 of a foot.

And then the length here
is 4 times 1/4 of a foot.

Well, that's equal to 4/4 of a
foot, which is equal to 1 foot.

So to figure out
the volume, we could

multiply the length times
the width times the height,

and these little dots here,
these aren't decimals.

I've written them
a little higher.

These are another way.

It's a shorthand
for multiplication,

instead of writing this
kind of x-looking thing,

this cross-looking thing.

So the length is 1.

The width is 1/2 of
a foot, so times 1/2.

And then the height
is another 1/2.

Let me do it this way.

The height is another 1/2, so
what's 1 times 1/2 times 1/2.

Well, that's going
to be equal to 1/4.

And this is a foot.

This is a foot.

This is a foot.

So foot times foot
times foot, that's

going to be feet to the
third power, or cubic feet.

1/4 of a cubic
foot, either way we

got the same result,
which is good.