### Video Transcript

In this video, we are going to look
at how to calculate the volume of a prism.

Now, there are two words in that
title that we need to be comfortable with. And the first of those is
volume. The volume of a 3D shape, or a
solid, remember, is the amount of space enclosed within that solid. Secondly, a prism, a prism is a
particular type of three-dimensional shape. And it’s one in which the cross
section of the solid is always the same. So, no matter where you cut the
solid along its length, you always see exactly the same face, exactly the same
shape, and exactly the same size.

Examples of prisms are things like
cubes and cuboids. Or cylinders are types of
prism. But you can also have prisms with
any shape at all as the cross section. So, you could have a triangular
prism, or a prism where the cross section is a parallelogram or a trapezium
perhaps. So, let’s look at an example of
calculating volume.

This question says, given that each
of the small cubes in the solid below has a volume of one cubic centimetre,
calculate the total volume of the solid.

So, if we know the volume of each
of the individual cubes, then really it’s just a question of adding up how many
cubes there are within this shape. But we don’t want to just go
through and add them all up because that would take a very long time if it was a
particularly large solid. So, we need to think about a
slightly quicker way of doing it.

So, if we look at the base of this
solid first of all, we can see that it has one, two, three, four, five, six cubes
arranged in a row in this direction. And it has two cubes in each row in
that direction. Which means that if I think about
just the number of cubes, or the volume, of just this bottom layer of the solid, so
the layer I have shaded in green, then it will be equal to six multiplied by
two. It will be 12 centimetres cubed, or
12 cubic centimetres.

But I don’t want the volume of just
one of these layers. I want the volume of this total
solid. So, I need to think about the fact
that there are, in fact, three layers of these. So, the total volume I need to
multiply is 12 by three. So, that gives me a volume of 36
cubic centimetres for the total solid.

Now, in the previous example, we
worked out the volume of that solid, which was a cuboid, just by counting how many
little cubes were within it. But that’s not always a
particularly practical way to answer such a question. So, let’s just think about a
general formula that we can use to calculate the volume of any prism.

So, I’ve chosen to use a triangular
prism for this. And we’ll just think about the
method behind what we did last time. Essentially what we did was we
worked out one layer of the prism, and then we multiplied it by how many there
were. So, what that’s equivalent to, in
terms of a general prism, is working out the area of the face on the front of the
prism, so this constant cross section. And then, multiplying it by the
height or the depth of the prism, which is essentially equivalent to how many layers
of this shape you have.

So, this gives us a method, or a
formula, for calculating the volume of any prism. We need to work out the area of its
cross section first of all, whether that’s a square, a rectangle, a triangle, a
circle, whatever it might be. And then, multiply it by the height
of this prism. And so, this formula will work
regardless of the type of prism that we’re given. So, let’s apply this to a couple of
questions.

The first question says, calculate
the volume of a cube of side length five centimetres.

So, in a cube, remember, all of the
sides are the same length cause all of the faces are squares. So, this is what we’re interested
in here, a cube with sides of length five. So, our formula tells us to work
out the area of the cross section. Now, in the case of a cube, the
cross section is a square. And the area of that is gonna be
five multiplied by five. And then, we have to multiply by
the height, or the depth, of the prism. And in the case of this cube,
that’s also five centimetres. So, our calculation for the volume
is five times five times five, or five cubed, which, of course, is 125 centimetres
cubed. Just a note about units, they are
cubic units because we’re talking about three-dimensional space, so centimetres
cubed, metres cubed, millimetres cubed, and so on.

Right, the next question asks us to
calculate the volume of the given cuboid. And we can see it’s got dimensions
of five centimetres, 12 centimetres, and three centimetres.

So, to calculate the volume, we
need the area of the cross section. And then, we need to multiply it by
the depth, or the height, of the prism. Now, a cuboid is quite special in
that whichever way you choose to slice it, whether it’s horizontal slices, vertical
slices, or left-to-right, that cross section that you see is always constant in any
direction. So, we can actually choose any of
these faces to be our cross section, but I’m gonna choose the front of the
prism. So, this face that I’ve marked in
green.

So, then, my calculation for the
volume, well, it’s the area of this cross section. So, it’s a rectangle with
dimensions of five and three. So, that’s gonna be five multiplied
by three. And then, I have to multiply by the
depth, or the height, of the prism, so that’s the remaining measurement of 12
centimetres. So, my calculation for this cuboid
is just five times three times 12. That gives me an answer then of 180
centimetres cubed for the volume of this cuboid.

So, our next question asks us to
calculate the volume of the triangular prism shown.

So, for this prism, that constant
cross section that you see wherever you slice it is a right-angled triangle. This triangle that’s on the front
face of the prism here. So, to calculate the volume,
remember, I need to do the area of the cross section. So, for the triangle, that’s base
times height over two, or three times four divided by two. And then, I need to multiply it by
the height, or the depth, of the prism, so that’s this measurement of eight
centimetres.

So, I have three times four over
two multiplied by eight. And if I then evaluate it, it gives
me an answer of 48 centimetres cubed for the volume of this prism. So, it is just a question of
thinking about what is the shape of the cross section of the prism you’re given. For a triangle, don’t forget to
divide by two. That is a common error that people
often make.

Right, our final question says,
calculate the volume of the prism below.

Now, if you look at this prism, you
will see that the front face, which is the constant cross section. So, this face that I’m shading in
in green here. This face is a trapezium because
it’s got one pair of parallel sides. And in fact, it’s a right-angled
trapezium. So, we’re going to calculate the
volume in exactly the same way, which means working out the area of this front face
and then multiplying by the depth. But we need to recall how to
calculate the area of a trapezium in order to do this.

So, to calculate the area of
trapezium, remember, you add together the parallel sides. So, in my case that’s this
10-centimetres side here and then this side, which is six centimetres. So, I’m gonna add them
together. And then, I halve that. What I’m essentially doing is
finding an average of that pair of parallel sides. Then, I need to multiply it by the
distance between the parallel sides. So, that’s this measure of seven
centimetres. So, that’s the first part of the
calculation. And all that’s doing is finding the
area of that trapezium, the front face of the prism.

Then, to turn it into a volume, I
need to multiply by the depth of the prism. So, I need to multiply by
eight. So, this gives me the full
calculation that I need. If I then go ahead and evaluate
that, then it gives me a volume of 448 cubic centimetres for this particular
prism.

So, as a reminder then, we’ve seen
that a prism is a 3D shape with a constant cross section, which means that if you
cut into it at any point you see exactly the same face. And in order to calculate the
volume of a prism, we work out the area of this constant cross section and then
multiply it by the height of the prism. Remember, we calculate the volume
for any type of prism in the same way. We just have to think about how we
calculate the area of the cross section each time depending what two-dimensional
shape it is. Whether it’s perhaps a square, a
rectangle, a cylinder, a trapezium, a parallelogram, or some other two-dimensional
shape.