### Video Transcript

In this video, we’ll learn how to
find volumes of triangular prisms and solve problems including real-life
situations. We’ll consider what we actually
mean by the word volume and how the properties of triangular prisms 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 rectangular prism or a cuboid,
for example, has a rectangular cross section. I could slice it down here or down
here, and the size and shape of that rectangle would stay the same throughout its
length. Similarly, a cylinder has a
circular cross section.

Now, in this video, we’re actually
interested in triangular prisms. These are, of course, prisms whose
cross section is a triangle. And in this video, we’re learning
how to calculate the volume of these shapes, where the volume is just the measure of
its total three-dimensional space. Let’s consider how we find the
volume of a cuboid or a rectangular prism and how that’s going to help us calculate
the volume of a triangular prism. Imagine we have a rectangular
prism. The dimensions of this prism are ℎ,
𝑤, and 𝑙 length units. We know that the formula that helps
us calculate the volume of such prism is ℎ times 𝑤 times 𝑙. And that will be in cubic
units.

Now, what we’re going to do is cut
our prism perfectly in half from here to here. In doing so, we create two
congruent, that means identical, triangular prisms. The volumes of each of these
triangular prisms must be exactly half of the volume of the rectangular prism. We could write that as ℎ times 𝑤
times 𝑙 over two or, alternatively, a half times ℎ times 𝑤 times 𝑙. But let’s consider the first part
of this formula, a half times ℎ times 𝑤. Do you recognize this formula? It looks a lot like the formula
that helps us calculate the area of a triangle, and that’s because it is. And we can therefore rewrite the
formula for the volume of a triangular prism. It’s the area of the triangular
cross section multiplied by its perpendicular length.

Now, in fact, we can generalize
this for any prism. Its volume is the area of its cross
section times its perpendicular length. Now, whilst we can use either of
these formulae, the second version is definitely the preferred form. Now, another way to think about
this is that the volume or the number of cubic units that a slice with a length of
one unit will hold will simply be the number of cubic units in that slice. Well, that’s just the same as the
area of the cross section. Then, the length tells us how many
of these slices we have, hence, area of the cross section times its length. Let’s begin with a simple
example.

Determine the volume of the given
triangular prism.

So, we’ve been given a triangular
prism with a number of dimensions. Remember, a prism is just a
three-dimensional shape with a constant cross section. And we know the formula that helps
us calculate the volume of a triangular prism is the area of its triangular cross
section multiplied by its length. Our cross section is in the shape
of a triangle, so we also recall the formula for the area of a triangle. It’s a half times base times
height. The base of our triangle measures
10 feet, whilst its height is six feet. So, its area is a half times 10
times six, which is 30.

Now, we’re working in feet, so the
area measurement will be in square feet. Let’s now substitute everything we
know into the formula for the volume of the prism. The area of its cross section is 30
while its perpendicular length, that’s the dimension that’s perpendicular to the
cross section, is 17. So, the volume is 30 multiplied by
17, which is 510. Once again, since we’re working in
feet, the volume will be cubic feet. And so, the volume of the given
triangular prism is 510 cubic feet.

We’ll now look at how to find the
volume of a triangular prism given information about the area of its cross
section.

A prism has a triangular cross
section whose area is 1.8 square meters. The length of the prism is 250
centimeters. Calculate the volume of the prism,
giving your answer in cubic meters.

We’re given some information about
a prism with a triangular cross section, in other words, a triangular prism. And we’re looking to calculate the
volume of this prism. So, let’s recall the formula we
need. The volume of a triangular prism is
found by multiplying the area of its cross section by its perpendicular length. Well, we see that the area of our
triangular cross section is 1.8 square meters. And the perpendicular length,
remember, that’s the dimension that’s perpendicular to the cross section, is 250
centimeters.

Notice that we have two different
units of measurement. We have square meters and
centimeters. And so, before we calculate the
volume, we either need to convert square meters into square centimeters or
centimeters into meters. The question is actually telling us
to give our answer in cubic meters, though. So, we’re going to convert the
length measurement into meters.

We know that there are 100
centimeters in a meter, so to convert 250 centimeters into meters, we’ll divide by
100. 250 divided by 100 is 2.5. So, the length of our prism is 2.5
meters. Note that, had we instead wanted to
convert our area into square centimeters, we would not have multiplied by 100 but by
100 squared. And so, we can see that whilst it’s
possible, it is actually much easier to convert the length measurements rather than
area or volume measurements.

We now substitute what we know
about our prism into the formula for its volume. Its area is 1.8, and its length is
2.5, so the volume is the product of these. But how do we calculate 1.8 times
2.5? We could, of course, use a
calculator, but let’s consider alternative methods. We’ll begin by calculating 18 times
25. There are a number of formal
written methods to do so, but let’s use the column method. We begin by multiplying the eight
by the five then the two, giving us 200 in the first row. We then multiply the one by the
five and the two, remembering of course we’re really multiplying these by 10. So, we add a zero here, giving us
250 in our second row. Finally, we add these two numbers,
giving us that 18 times 25 is 450.

So, how does this help us calculate
1.8 times 2.5? Well, 18 is 10 times larger than
1.8, as is 25 to 2.5. This means our answer, 450, is
going to be 10 times and then another 10 times bigger or 100 times bigger. So, to find the value of 1.8 times
2.5, we’re going to divide by 100. 450 divided by 100 is 4.5. And remember, our answer is in
cubic meters. And so, the volume of our
triangular prism is 4.5 cubic meters.

In our next example, we’ll look at
how we can find the unknown length of a triangular prism when given information
about its volume.

The diagram shows a triangular
prism whose volume is 1,932 cubic centimeters. Calculate the value of 𝑥.

We’ve been given some information
about the volume of a triangular prism, so let’s begin by recalling the formula for
this. The volume of a triangular prism is
the area of the triangular cross section multiplied by its perpendicular length. We’re told that the volume is 1,932
cubic centimeters. We also know that the length that’s
perpendicular to the cross section is 14 centimeters. So, we can therefore say that 1,932
must be equal to the area of the triangle multiplied by 14. Let’s solve this equation to find
the area of the triangle in square centimeters.

We’ll divide through by 14. That tells us that the area of the
triangle is 1,932 divided by 14 which is 138 or 138 square centimeters. Let’s now clear some space and
remind ourselves on the formula for the area of a triangle. The area of a triangle is a half
multiplied by the length of its base multiplied by its height. Now, it doesn’t actually matter
that our triangle is scalene. We use the same formula. Remember, we calculated the area of
our triangular cross section to be 138 square centimeters. The base is 𝑥 centimeters, and the
height is 23 centimeters. So, we have another equation. 138 is a half times 𝑥 times
23.

This time, we’ll begin to solve
this equation by multiplying both sides by two. 138 times two is 276. So, our equation becomes 276 equals
𝑥 times 23 or 23𝑥. Finally, we’re going to divide both
sides of this equation by 23. So, 𝑥 is 276 divided by 23, which
is 12. And so, given the information about
our triangular prism and its volume, we find that 𝑥 in our diagram is equal to
12.

Which of the following has the
largest volume?

We’ve been given four prisms. Remember, a prism is a shape which
has a constant cross section. In order to compare their volumes,
we’re going to begin by calculating the volume of each. And so, we recall that to find the
volume of a rectangular prism and, in fact, a cube, we simply multiply its three
dimensions. Our first shape is a cube, so its
three dimensions are seven inches, seven inches, and seven inches. Its volume, therefore, must be
seven times seven times seven or seven cubed. Seven cubed is 343. And since we’re working in inches,
the volume of our first shape is 343 cubic inches.

Similarly, the volume of this
rectangular prism is eight times six times five. We remember that multiplication is
commutative. It can be performed in any
order. Six times five is 30, so we need to
work out eight times 30. And since eight times three is 24,
we see that eight times 30 is 240. And the volume of our second shape
is 240 cubic inches. But what about the triangular
prisms? Well, the formula we tend to use is
that it’s the area of the triangular cross section times its length, where the
length is the length of the dimension perpendicular to the cross section. We also know that the area of a
triangle is found by multiplying its base by its height and then dividing by two or
a half base times height.

In our first prism, the one whose
cross section is a scalene triangle, the length of its base is 12 inches and its
height is nine inches. And so, the area of this triangle
is a half times 12 times nine, which is 54 or 54 square inches. The length of the dimension
perpendicular to this cross section is eight inches, so the volume of the shape is
54 times eight, which is 432 cubic inches.

Let’s repeat this process for our
fourth prism. This time its cross section is a
right-angle triangle, and the length of its base is 18, and its height is four. So, the area of this triangle is a
half times 18 times four, which is 36 or 36 square inches. The length of the dimension
perpendicular to this face is 10, so the volume of the shape is 36 times 10, which
is 360. So, our final shape has a volume of
360 cubic inches. And we can say then that the shape
which has the largest volume is the triangular prism whose cross section is a
scalene triangle.

In our final example, we’ll look at
some real-life applications of the volume of a triangular prism.

A vase in the shape of a triangular
prism is standing on its triangular face as shown in the diagram. 200 cubic centimeters of water is
poured into the vase. A bunch of flowers is then placed
into the vase, causing the height of the water to rise by 1.8 centimeters. How much more water can be added to
the vase until it is completely full? Give your answer in cubic
centimeters.

So firstly, we note we are dealing
with a triangular prism. It’s a prism whose cross section is
in the shape of a triangle. We’re told that 200 cubic
centimeters of water is initially poured into the vase, but that a bunch of flowers
is placed in, which causes the height of the water to rise by 1.8 centimeters. We want to work out how much more
space is in the vase essentially. So, we’re going to need to perform
this in a few steps. Let’s begin by working out the
height that the water reaches when 200 centimeters cubed is poured into the
vase.

We begin by recalling that the
volume of a triangular prism is found by multiplying the area of its triangular
cross section by its length or its height. 200 cubic centimeters of water is
poured into the vase, so the volume of our water is 200. We know the area of a triangle is a
half multiplied by its base multiplied by its height. Well, the base of our triangle is
five, and its height is eight, so its area is a half times five times eight, which
is 20 square centimeters. We’re trying to work out the
height, let’s call that ℎ, that the water reaches in the vase. So, we have an equation: 200 equals
20 times ℎ.

Let’s find the value of ℎ by
dividing both sides of this equation by 20. ℎ is therefore 200 divided by 20,
which is 10 or 10 centimeters. We can say then that the initial
height of water in the vase is 10 centimeters. We’re now going to work out the
height of water in the vase when the flowers are added. So, that must be 10 plus 1.8, since
it rises by 1.8. And we can now say that ℎ sub two,
the second height of water, is 11.8 centimeters. We essentially want to work out the
remaining volume in the vase. So, we’re going to find the
difference between the total height of our vase and the height that the water now
reaches. That’s 14 minus 11.8, which is 2.2
centimeters.

So, we need to work out the volume
of the slice of the vase, which is 2.2 centimeters high. Remember, we already calculated the
area of the triangular cross section to be 20. And the whole point of a prism is
that its cross section doesn’t change. So, the area of the cross section
we’re interested in is also 20. And the volume is 20 times 2.2,
which is 44 or 44 cubic centimeters. And so, 44 cubic centimeters of
water can be added to the vase to make it completely full.

Now, interestingly, this wasn’t the
only way we could’ve answered this problem. Instead, we could’ve worked out the
volume of this slice. And in fact, had we done so, we
would’ve found that the volume of this slice is 36 cubic centimeters, meaning that
after the flowers are added, we have a total of 200 plus 36. That’s 236 cubic centimeters of
volume being used. We would have then had to have
calculated the volume of the entire vase. Had we done that, we would have
found it to be 280 or 280 cubic centimeters. The amount of space left in the
vase is 280 minus 236 which is, once again, 44 cubic centimeters.

In this video, we’ve learned that a
prism is a three-dimensional shape with a constant cross section. We saw that we can find the volume
of a triangular prism by multiplying the area of its triangular cross section by its
length and that we need to be careful to ensure that our units match before we
perform any calculations.