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 twelve centimetres cubed or twelve cubic
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 twelve by three. So that gives me a volume of thirty-six 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 were 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 because 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 a hundred and twenty-five 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.
Well the next question asks us to calculate the volume of the given cuboid, and
we can see it’s got dimensions of five centimetres, twelve 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 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 twelve centimetres.
So my calculation for this cuboid is just five times three times twelve. That gives me an answer then of one hundred and eighty 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 forty-eight
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.
Well our final question says calculate the volume of the prism below. Now if you
look at this prism, you will see 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 ten 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 four hundred and forty-eight 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.