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# Measuring volume as area times length

Video transcript

I have this figure here. You could call it a
rectangular prism. And I want to
measure its volume. And I'm defining my unit cube
as being a 1 centimeter by 1 centimeter by 1 centimeter cube. It has 1 centimeter width, 1
centimeter depth, 1 centimeter height. And I will call this, this is
equal to 1 cubic centimeter. So I want to measure this volume
in terms of cubic centimeters. We've already seen that we
can do that by saying, hey, how many of these
cubic centimeters can fit into this figure without
them overlapping in any way? So if we had this in our
hands, we could kind of try to go around it
and try to count it, but it's hard to see
here because there's some cubes that we can't
see behind the ones that we are seeing. So I'm going to try
different tactics at it. So first, let's just think
about what we can observe. So we see that this
one, if we measure its different
dimensions, its width, it's 2 of the unit length wide. So it's 2 centimeters wide. It's 4 of our unit length--
we're defining our unit length as a centimeter-- it's 4
of our unit length high. So this dimension right
over here is 4 centimeters. And it is 3 of our
unit length deep. So this dimension right
over here is 3 centimeters. So I want to explore
if we can somehow use these numbers
to figure out how many of these cubic centimeters
would fit into this figure. And the first way I'm
going to think about it is by looking at slices. So I'm going to take
this slice right over here of our
original figure. And let's think about
how using these numbers, we can figure out how many
unit cubes were in that slice. Well this is 2 centimeters wide
and it is 4 centimeters high. And you might be
saying, hey Sal, I could just count these things. I could get 8 squares here. But what if there
was a ton there? It would be a lot harder. And you might realize
well I could just multiply the width times the
height, that would give me the area of this
surface right over here. And it's only 1
deep so that also will give me the
number of cubes. So let's do that. Let's find the area here. Well that's going to be
2 centimeters times 4 centimeters. That gives us the area of this. And then if we want to find
out the number of cubes, well that's also going to
be equivalent to the number of cubes. So we have 8 square
centimeters is this area, and the number of cubes is 8. And if we want the number
of cubes in the whole thing, we just have to multiply
by the number of slices. And we see that we need
one, two, three slices. This is 3 centimeters deep. So we're going to
multiply that times 3. So we took the area
of one surface. We took the area of this
surface right over here. And then we multiply by the
depth, that essentially gives us the number of cubes because
the area of this surface gives us the number of cubes in
an slice that is 1 cube deep. And then we would have to
have 3 slices like that. So we would have to
have this is 1 slice. We would have to have another
slice, another slice and then another slice in order to
construct the original figure. So 2 centimeters times 4
centimeters times 3 centimeters would give us our volume. Let's see if that works out. 2 times 4 is 8 times 3 is 24. Let me do that in
that pink color. 24 centimeters cubed, or I
could say cubic centimeters. So that's one way to
measure the volume. Now there's multiple
surfaces here. I happened to pick
this surface, but I could have picked another one. I could have picked this
surface right over here and done the exact same thing. So let's pick this surface
and do the exact same thing. This surface is 3
centimeters by 4 centimeters. Let me do that in
that blue color. Color changing is
always difficult. So its area is going to
be 12 square centimeters is the area of this surface. And 12 is also the
number of cubes that we have in that slice. And so how many slices do
we need like this in order to construct the
original figure? Well we need, it's
2 centimeters deep. This is only 1 centimeter
deep so we need two of them to construct the
original figure. So we can essentially find
the area of that first surface which was 3 times
4, and then multiply that times the width, times how
many of those slices you need, so times 2. And once again, this is
going to be 3 times 4 is 12 times 2 is 24. I didn't write the
units this first time. But that's going to
give us the count of how many cubic
centimeters we have, how many unit cubes we can fit. So once again, this is
24 cubic centimeters. And you could imagine, you
could do the same thing, not with this surface,
not with this surface, but with the top surface. The top surface is
3 centimeters deep. And 2 centimeters wide. So you could view
its area or its area is going to be 3 centimeters
times 2 centimeters. So that area is-- let me do
it in the same colors-- 3 centimeters times 2 centimeters
which is 6 square centimeters. And that also tells
you that there's going to be 6 cubes in
this one cube deep slice. But how many of these
slices do you need? Well you have this whole
thing is 4 centimeters tall, and this thing is
only 1 centimeter so you're going to
need four of them. So that's 2, 3, try to draw
it as neatly as I can, and 4. You're going to need 4 of these. So to figure out
the whole volume, you going to have to
take that and multiply that times 4 centimeters. So once again, 3 times 2 is
6 square centimeters times 4 centimeters is 24
cubic centimeters. So it doesn't matter what
order you multiply these in. You could view this and
take the area of one side then multiply it
times the depth. Or you could take the surface
area of another height and multiply it times the height
or the width or the depth. And these are all the scenarios. But what it shows is that it
doesn't matter what order we multiply these
three dimensions in. You could take the 2
times the 4 first and then multiply it by the 3. Or you take the 3 times the
4 first and then multiply it by the 2. And or you could take
the 2 times the 3 first, and then multiply by the 4. When you're
multiplying, it doesn't matter what order
you're doing these in. And so if you have a
rectangular prism like this and you know it's
three dimensions, you know it's 2 centimeters
wide, 3 centimeters deep, and 4 centimeters tall,
you could say, hey, the volume of this thing,
the number of unit cubes, the number of cubic
centimeters it can fit is going to be 2 centimeters
times 4 centimeters times 3 centimeters, which
we've seen three times already is 24 cubic centimeters.