Video Transcript
In this video, we’ll learn how to
calculate volumes of rectangular prisms and cubes, given their dimensions, and solve
problems, including real-life situations. We’ll consider what we actually
mean by the word “volume” and how the properties of rectangular prisms and cubes can
help us to derive and use a formula for their volume.
So, our first question is, what is
a prism? A prism is a three-dimensional
shape with a constant cross section. In other words, the cross section
has the same shape and size throughout its length. A triangular prism, for example,
has a triangular cross section. I could slice down here or down
here, and the size and shape of that triangle would stay the same. Similarly, a cylinder has a
constant cross section; it’s a circle. Now, we’re interested in
rectangular prisms, like this, and cubes. A cube is simply a rectangular
prism whose dimensions are all the same. We notice its faces — that’s the
flat surfaces of the shape — are all squares.
Now, in this video, we’re learning
how to calculate the volume of these shapes. The volume of a shape is a measure
of its total three-dimensional space. And the easiest way to measure the
three-dimensional space is to consider how many cubic units a shape contains. A cubic unit is simply a cube that
measures one unit by one unit by one unit. Let’s have a look at an example of
this.
Find the volume of the cuboid.
The volume of a cuboid or a
rectangular prism is a measure of its three-dimensional space. We can work out the volume by
considering how many unit cubes the shape contains. In this case, we see that our
cuboid is split into cubes which measure one centimeter by one centimeter by one
centimeter. That’s one cubic centimeter. And one method we have is to simply
count them, beginning with the front of our shape. Here, we have one, two, three,
four, five, six, seven, eight, nine, 10, 11, 12, 13, 14, 15 cubes. But is there a quicker way to
calculate this? Well, yes! We have five rows, each containing
three cubes. So, we could have simply multiplied
five and three to see that there are 15 cubes on the front of our shape.
We now look at the depth of our
shape. It’s two centimeters. This means we’ve essentially got
two identical slices, each containing 15 cubes. The volume is, therefore, 15
multiplied by two, which is 30. Now, each cube is one centimeter
cubed. So, the total volume of our cuboid
is 30 cubic centimeters. But is there a quicker way we could
have done this? Well, yes. Rather than counting cubes, we saw
that we can multiply five by three, and then we multiply this by two.
In other words, we can multiply the
width by the height by the length, and that’s the formula for the volume of a
rectangular prism or a cuboid. It’s the product of its length, its
width, and its height. In this case, that was five times
three times two. But remember, multiplication is
commutative. So, we can actually do this in any
order and get the same answer of 30 cubic centimeters.
Let’s now have a look at how we can
use this formula to solve problems involving the volume of rectangular prisms and
cubes.
Which of the following describes
how the volume of a rectangular prism is affected after doubling all three
dimensions? Is it (A) 𝑉 sub new is equal to
six times 𝑉 sub old? (B) 𝑉 new is equal to 𝑉 old
squared. (C) 𝑉 new is equal to two times 𝑉
old. (D) 𝑉 new is equal to four times
𝑉 old. Or (E) 𝑉 new is equal to eight
times 𝑉 old.
To answer this question, we’ll
begin by recalling the formula for the volume of a rectangular prism. The volume 𝑉 of a rectangular
prism is the product of its length, its width, and its height. So, let’s call the volume of our
original shape 𝑉 old. It’s 𝑤𝑙ℎ. Now, of course, we could do this in
any order. So, we could write 𝑙𝑤ℎ or any
other combination. We’re now going to take our
original rectangular prism and double all of its dimensions.
The height of our new shape is two
times ℎ, which is two ℎ. The width is now two times 𝑤. That’s two 𝑤. And the length is two times 𝑙. That’s two 𝑙. And so, we can now calculate the
volume of the new shape. It’s still the product of all of
its dimensions, but this time that’s two 𝑤 times two 𝑙 times two ℎ. When we multiply algebraic
expressions, such as this, we begin by multiplying the numbers. And so, two times two times two is
eight. And the new volume is eight
𝑤𝑙ℎ.
We now compare the original volume
to the new volume. And since the original volume is
𝑤𝑙ℎ and the new volume is eight times this, this must mean that the new volume is
eight times the old volume. The correct answer, therefore, is
(E) 𝑉 sub new is equal to eight times 𝑉 sub old.
We’ll now consider how we can use
the formula for the volume of a rectangular prism within a real-world context.
A man needs to store 16,170 cubic
centimeters of rice in a container. He has one box which is a cuboid
with dimensions of 35 centimeters, 22 centimeters, and 21 centimeters and another
box which is a cube with length 22 centimeters. Which box should he use?
Remember a measure of capacity or
the amount of space a three-dimensional shape holds is volume. And the formula for the volume of a
rectangular prism or a cuboid is width times height times length. Essentially, it’s the product of
its three dimensions. Now, we can calculate this product
in any order. We also have a cube in this
question, but a cube is simply a cuboid whose dimensions are all equal. And so, the volume of a cube is
simply the length of one of its edges cubed. So, we’ll begin by calculating the
volume of each shape.
The dimensions of the cuboid are 35
centimeters, 22 centimeters, and 21 centimeters. So, the volume is 35 times 22 times
21, which is 16,170. Now, we’re multiplying centimeters
by centimeters by centimeters. And so, the units here are cubic
centimeters or centimeters cubed. Now, in fact, this is the exact
same value as the amount of rice that the man needs to store. But let’s double-check with the
volume of the cube.
This time, the length of each edge
is 22 centimeters. And so, the volume of our cube is
22 times 22 times 22 or 22 cubed. That gives us a volume of 10,648
cubic centimeters. This is indeed going to be too
small, so he should use the cuboid to store the rice.
In our next question, we’ll
consider how to calculate the volume of a rectangular prism given the area of one of
its faces.
Cuboid 𝐴 has dimensions of 56
centimeters, 40 centimeters, and 34 centimeters. Cuboid 𝐵 has a base area of 2,904
square centimeters and a height of 36 centimeters. Which cuboid is greater in
volume?
We begin by recalling the formula
for the volume of a rectangular prism or a cuboid. It’s the product of its three
dimensions. We could write that as width times
height times length in any order. And so, we can calculate the volume
of cuboid 𝐴 fairly easily. It’s 56 multiplied by 40 multiplied
by 34, which is 76,160. Our units are centimeters, so the
units for the volume are cubic centimeters or centimeters cubed. But what about the volume of cuboid
𝐵?
Well, we’ll go back to our formula
for the volume of a cuboid. And since multiplication is
commutative, we know we can do it in any order. So, we can rewrite this as width
times length multiplied by height. But, of course, width times length
gives us the area of a rectangle. In this case, that’s the area of
the base of our cuboid. And so, we can alternatively say
that the volume of a cuboid is equal to the area of its base multiplied by its
height, where the height is the side that’s perpendicular to the base.
We also sometimes say that the
volume of a cuboid is equal to the area of its cross section multiplied by its
length or its height. We can, therefore, calculate the
volume of cuboid 𝐵 by multiplying 2,904 — that’s the area of its base — by its
height; that’s 36. That gives us a value of 104,544
cubic centimeters. We can quite clearly see that
76,160 is less than 104,544, meaning that cuboid 𝐵 is greater in volume.
We’ll now have a look at how we can
use information about the volume to solve problems in a real-world context.
Given that 405 cubic centimeters of
water is poured into a rectangular-prism-shaped vessel with a square base whose side
length is nine centimeters, find the height of water in the vessel.
In his question, we’ve been given
some information about the volume of water being poured into a
rectangular-prism-shaped vessel. This vessel has a square base with
side length of nine centimeters. So let’s sketch this out. Here is this vessel. Now, we don’t know what the height
of the water is in the vessel when it’s poured in. So, let’s call that ℎ
centimeters. We do know that the amount of space
this takes up in three dimensions is 405 cubic centimeters. And we also know that this is the
volume. And the volume of a cuboid is equal
to the area of its base multiplied by its perpendicular height.
Now, we can calculate the area of
the base of our vessel. It’s simply a square. So, its area is nine multiplied by
nine, which is 81. We’re working in centimeters. So, the area of the base of our
vessel is 81 square centimeters. We, in fact, also know that the
volume of our water is 405 cubic centimeters. And we’ve said that its height is
equal to ℎ.
We can, therefore, form an equation
in ℎ. We can say that 405, remember,
that’s the volume, is equal to the area of the base, that’s 81, times ℎ or 405
equals 81ℎ. We want to solve for ℎ. So, we’re going to divide both
sides of our equation by 81. That gives us ℎ is equal to
five. And we can, therefore, say that the
height of water in the vessel is five centimeters.
In our very final example, we’ll
look at how we need to be extra careful when dealing with different units in our
question.
Find the volume of the cuboid.
We recall that the volume of a
cuboid or a rectangular prism is the product of its three dimensions. So, it’s its width multiplied by
its length multiplied by its height. Now, we need to be extra careful
here. We are given that the width of our
cuboid is 0.5 meters and its length is four meters. Its height, however, is 85
centimeters. The units are different to the
other dimensions. And so, before we calculate the
volume, we’re going to ensure that all our units are the same.
Now, we could do this in one of two
ways. We could convert all of our
measurements to centimeters and give our volume in cubic centimeters or convert our
measurements to meters and give our volume in cubic meters. We’ll have a look at both
examples. We know that there are 100
centimeters in a meter. And so, to convert from meters to
centimeters, we multiply by 100. 0.5 meters is 0.5 times 100, which
is 50 centimeters. Similarly, four meters is four
times 100, which is 400 centimeters. The volume, therefore, of our
cuboid in cubic centimeters is 50 times 400 times 85, which is 1,700,000 cubic
centimeters.
And now, let’s see what happens
when we calculate the volume in cubic meters. This time, to change from
centimeters into meters, we’re going to divide by 100. So, 85 centimeters is 85 divided by
100, which is 0.85 meters. In cubic meters then, our volume is
0.5 times four times 0.85, which gives us a volume of 1.7 cubic meters. And so, we’ve calculated the volume
of our cuboid in both cubic centimeters and cubic meters.
Now, a common mistake is to think
that to convert between cubic centimeters and cubic meters, we multiply or divide by
100. We can see quite clearly that
that’s not the case here. In fact, to convert from cubic
centimeters to cubic meters, we divide by 100 cubed. And to convert the other way, we
multiply by 100 cubed.
In this video, we’ve learned that a
rectangular prism is a solid shape that looks a little bit like a box. It has six rectangular faces, and
we call its dimensions length, width, and height. We learned that the volume of this
prism is the product of these dimensions. It’s width times length times
height. And we can calculate this in any
order.
We also saw that we can
alternatively say that the volume of a rectangular prism is equal to the area of its
base or its cross section multiplied by its height, where the height is the
dimension perpendicular to the base. And remember, we should always
check that dimensions are given in the same unit before working out volume of a
three-dimensional shape.
