Video Transcript
In this video, we’re going to look
at how to calculate the volume of oblique prisms.
So, first we need to be clear what
is meant by the term oblique prism. And there’s a diagram on the screen
here to help us understand this. You’ll perhaps already be familiar
with what’s called a right prism, which I’ve drawn on the left of the screen. And what you’ll notice about this
is it for the right prism, the lateral faces are perpendicular to the bases. So, I’ve marked one of the lateral
faces in orange and then the base in green. And you can see that they are
perpendicular to each other, or at a right angle, which is where the name right
prism comes from.
Now, if you look at the oblique
prism, and if I do the same shading again, so there’s the base marked in green and
one of the lateral faces shaded in orange, you’ll see that this time they are not
perpendicular to each other. So, there’s the difference between
these two types of prism. In the right prism, as I said, the
lateral faces are perpendicular to the bases, whereas in the oblique prism that is
not the case.
What this also means that in a
right prism the lateral faces are rectangles, whereas in an oblique prism the
lateral faces are parallelograms. Now, you’ll have already seen how
to calculate the volume of a right prism. This video we’re focusing on the
oblique prisms. So, in order to think about the
volume of oblique prisms, we need to rely on a principle called Cavalieri’s
principle. And it says the following. If two solids have the same height
and the same cross-sectional area at every level, then they have the same
volume.
So, take a look at the diagram. These two prisms, one is a right
prism, and one is an oblique prism. They have the same base area marked
in green. And because they’re prisms, they
will have that same area at every level throughout their height. And they also have the same height
marked in ℎ. Now, an important thing to know is
that it is the perpendicular height. So, in the case of the right prism,
that’s just its usual height. And in the case of the oblique
prism, it’s that height that’s perpendicular to the base. You’ll see I’ve drawn a right angle
in a continuation of that baseline there. So, Cavalieri’s principle tells us
that the volume of these two prisms are equal.
One way perhaps to help visualize
this is to think about a stack of coins. So, in one case, we have these
coins stacked directly on top of each other like in the right prism. Whereas in the other, we have them
in a sort of diagonal stack if you could get them to balance like that, like in an
oblique prism. And if you were to do this
yourself, you’d see that the height of those two stacks of coins is the same. And of course, the volume is the
same because they’re just the same coins arranged in a different formation. So, that gives a helpful physical
demonstration of this principle.
What all of this means then, in the
case of calculating the volume of an oblique prism is we can essentially treat them
exactly the same as we do right prisms by working out the cross-sectional area and
then multiplying it by the height of the prism. So, we can treat these two types of
prisms in exactly the same way. The only caveat with the oblique
prism is we need to make sure we’re using the perpendicular height and not a sloping
height.
So, let’s look at applying this to
our first question.
We’re asked to calculate the volume
of the oblique rectangular prism below.
So, remember from the previous
discussion then that the volume of this oblique prism will be the area of the cross
section, or the base, multiplied by the height. So, I’m going to use 𝐵 to
represent base and ℎ to represent height in my formulae here. So, looking at the prism then, the
base of it, or in this case the top that I’ve shaded in, is a rectangle with
dimensions of two and five. So, that will be fine to work out
its area.
We then just need to think about
the height of the prism carefully. Because we’ve actually been given
two different heights. We’ve been given the six meters,
which is that sloping height. And we’ve been given four meters,
which is the perpendicular height and, remember, is the perpendicular height that we
need. So, in this question, we’ve
actually been given more information than is necessary in order to test that we
truly understand the method for calculation a volume of an oblique prism.
So, our calculation then, the
volume is the base area. Well, as we said that’s a rectangle
with dimensions of two and five, so two times five. Multiplied by the height, and we
must use that measurement of four meters, the perpendicular height. So, working this out gives us a
volume of 40 cubic meters for this oblique rectangular prism.
Okay, the next question says,
calculate the volume of an oblique hexagonal prism with a perpendicular height of 10
centimetres and a base area of 65 centimetres squared.
So, let’s recall the volume formula
that we need. And of course, it’s this formula
that the volume is equal to the base multiplied by the perpendicular height. So, we’ve been given both of those
measurements. We just need to substitute them
into this formula. So, in the case of this hexagonal
prism, the base area is 65, the perpendicular height is 10. So, to calculate the volume, we’re
multiplying 65 by 10. And this gives us an answer then of
650 cubic centimetres.
The next question asks us to find
the volume of an oblique square prism with height 7.2 millimetres and base edges of
length 4.5 millimetres.
So, as always, we need to recall
that volume formula, which is the base area multiplied by the height. Now, we’ve got the height. It’s 7.2 millimetres. And because this is an oblique
square prism, we can calculate the area of the base by multiplying its two sides
together, so 4.5 times 4.5. So, our calculation of the volume
then is just 4.5 multiplied by 4.5 to give the base area then multiplied by 7.2,
which is the height. This gives us an answer then of
145.8 millimetres cubed.
So, in each of these questions, the
only real consideration so far has been the shape of the base because, of course,
that affects the calculation that we do in order to find its area. In the case of a square or
rectangle, it’s relatively straightforward. Remember, of course, if it’s a
triangle, you have to divide by two. Or if it’s another type of
two-dimensional shape, you just have to record the relevant formula for calculating
its area.
Right, the final question asks us
to find the volume of the oblique cylinder shown.
So, we need, of course, our volume
formula. And in the case of the cylinder,
the base is, of course, a circle. So, we also need to recall the
formula for finding the area of circle, which remember is 𝜋𝑟 squared, where 𝑟
represents the radius of the circle. So, using those two formula, let’s
calculate the volume of this oblique cylinder.
So, base area, first of all, is
𝜋𝑟 squared. Well, if we look at the diagram, we
haven’t actually been given the radius. We’ve been given the diameter of
the circle, which is six centimetres. So, we need to halve it in order to
find the radius. So, we have 𝜋 multiplied by three
squared for the area of the base. Then, we need to multiply by the
height, so multiplied by five. This gives us an answer then of
45𝜋. And we could leave our answer like
that if we didn’t have a calculator, or we wanted an exact answer, or indeed if it
was requested. But I’ll go on and evaluate this
answer as a decimal. And this gives me an answer then of
141.4 cubic centimetres. And that’s been rounded to one
decimal place.
So, to summarize then, when working
with oblique prisms due to Cavalieri’s principle, you can treat them in the same way
that you do right prisms. And you can calculate their volumes
by working out the area of the base and then multiplying by the height. Just make sure you are using the
perpendicular height as opposed to any kind of slant height of the prism.
