In this video, we are going to look at some more advanced problems related to
calculating the surface area and the volume of prisms. So this is the first problem; it says the surface area of a cube is two hundred
and ninety-four centimeters squared. We’re then asked to calculate the volume of this cube.
Now in order to calculate the volume of a cube, we would need to know what the
length of its sides are. However, we aren’t given that; we’re given the surface area instead. So
this problem is going to involve working backwards from knowing the surface area to
calculating the side length and then calculating the volume of the cube.
So let’s think about how to approach this problem then. I’ve sketched myself a
cube and have decided to call this unknown side length 𝑥. So in order to work out the total surface area, what would have been done is we’d
have worked out the area of each of these faces. Now they’re all squares, and as the
dimensions are 𝑥, the area of each face will be 𝑥 squared.
Now on a cube there is six faces, six identical faces, so the total surface area of
this cube would be six 𝑥 squared. And we’re told in the question that this is equal to two
hundred and ninety-four centimeters squared. So what this enables me to do is set up an
equation involving this unknown side length 𝑥.
So now I need to solve this equation in order to work out what 𝑥 is. Well
the first step would be to divide both sides of this equation by six, and that will give me the 𝑥 squared is equal to forty-nine. And in the next
step to work what 𝑥 is, I need to take the square root of both sides of this equation.
And that tells me that 𝑥, the side length of this cube, is seven centimeters. Okay so now I know the side length; it’s relatively straightforward to calculate
the volume of this cube. I just need to remember that to calculate the volume of a cube and
multiply these three dimensions together, so I’m doing 𝑥 times 𝑥 times 𝑥 or 𝑥 cubed.
So the volume of this cube is seven times seven times seven, which gives me an answer of three hundred and forty-three centimeters cubed or
cubic centimeters for the volume of the cube.
So in that question, we still have to work backwards from knowing the surface
area of the cube, calculating the side length, and then calculating the volume. A similar
question could perhaps give you the volume and ask you to calculate the surface area.
So here is such a question; it says the total surface area of the cuboid shown is
eighty-six centimeters squared, and then we’re asked to calculate the volume of this cuboid. So what you’ll notice looking at the diagram is we’ve been given two of the
dimensions, three centimeters and five centimeters, but we’re not told what that third dimension is. So it’s
been given the letter 𝑦.
In order to calculate the volume, we’re going to need to work out what 𝑦 is. So
we’re gonna need to work backwards from knowing the total surface area, calculating this
missing dimension 𝑦, and then using it in order to calculate the volume.
So let’s think about the different faces of the cuboid that would make up this
total surface area. So we have the front and the back of the cuboid, so we have these faces here.
Now if you think about the dimensions of those faces, they are three centimeters and five
centimeters, so the area of each of those faces is fifteen centimeters squared.
Now of course, there are two of them, the front and the back, so we’ll have to
remember that later on in our method. Now let’s think about the top and the base of this cuboid, so this face on the
top and the one on the bottom. Now the dimensions of that are three centimeters and then this unknown
measurement 𝑦, which means the area of those faces; well it’s three multiplied by 𝑦, so it’ll
be three 𝑦 centimeters squared. Again remember there are two of those.
Finally, let’s think about the sides of this cuboid, so the ones I’m
currently shading in orange, and the dimensions of those were the height is five and then the
width of this rectangle is again the unknown measurement 𝑦, so it’d be five multiplied by 𝑦
or five 𝑦 centimeters squared.
So now we have all the information we need to formulate an equation for this
surface area. So we would have added all of these together, so we’d have done three 𝑦 plus five
𝑦 plus fifteen. But remember there are two of each of those, so we’d also have multiplied that
whole expression by two.
And then the answer to that we’re told is eighty-six. So what we’ve done here is formulate an equation for the total surface area in terms
of that unknown measurement 𝑦.
We now want to solve this equation, so let’s tidy up a little bit first. Three
𝑦 plus five 𝑦 is eight 𝑦 in the brackets. So I have two lots of eight 𝑦 plus fifteen. Next I’m gonna divide both sides
of this equation by two.
So that gives me eight 𝑦 plus fifteen is equal to forty-three. Now I’m gonna
subtract fifteen from both sides of this equation. And that would give me eight 𝑦 is equal to twenty-eight. The final step in
calculating 𝑦 is I need to divide both sides of the equation by eight.
And that tells me that 𝑦 is equal to three point five. So we’ve worked out that this
missing dimension of the cuboid is equal to three point five centimeters. Now that I know that, calculating the volume is relatively straightforward. I need
to multiply the three dimensions of the cuboid together. So I’m gonna be doing three
multiplied by five multiplied by three point five.
And if I then evaluate that, it gives me an answer of fifty-two point five
centimeters cubed. So in this question, we knew the total surface area. In order to calculate the
volume, we had to work out the unknown length in this cuboid. We did that by formulating an
algebraic equation and then solving it. Once we had that, we could just calculate the volume of
the cuboid in the usual way.
Okay the next question we’re going to look at says, Tom wants to pour fifty-five
liters of water into the drinking trough shown. Will all of the liquid fit? So looking at the diagram, we have a drinking trough in the shape of a
triangular prism, and we want to know the capacity of that trough. Is it more or less than
So we’re going to need to work out the volume of this trough. Now there’s
something we need to be careful of, which is if you look at the diagram, the units that we’ve
been given are mixed: two of the measurements in centimeters and one of the units is meters. So
we need to make sure they’re all in the same unit before we start trying to calculate this volume.
So the simplest thing to do then is to convert this one point five meters into a
hundred and fifty centimeters, and we’ll use that in our calculations. So we’re going to calculate the volume of this triangular prism, this trough. Now
that answer will be in centimeters cubed, so we’ll have to think later about how that relates
to liters, because the information in the question is about fifty-five liters of water.
So for the volume in this triangular prism then, remember we need to work out the
area of the cross section, so that’s this triangle here, and then multiply it by the height or
the depth of the prism, so that’s the hundred and fifty centimeters.
Now the cross section is a triangle, so we’re going to be doing base times
perpendicular height over two, so that’s thirty-five times twenty over two. And then we’re going to multiply by the depth of the prism, so that’s the one
hundred and fifty centimeters.
So this gives us a volume of fifty-two thousand five hundred centimeters cubed
for the drinking trough. So we need to think how this relates to liters and we need to record a key
piece of information, which is that one centimeter cubed is equivalent to one milliliter.
So what this tells us then is that the volume of this trough in terms of how
much liquid can be poured into it; well it’s a one-for-one conversion between centimeters cubed
and milliliters, so we can fit fifty-two thousand five hundred milliliters of water in this
Now the information in the question is given in terms of liters, so let’s convert
this to liters by dividing by one thousand. And this tells us then that the volume of this trough is fifty-two point five liters.
So Tom wants to pour fifty-five liters in there; well he’s not gonna be able to; he’s gonna have
some liquid that overflows from the trough. So in answer to the question will all the
liquid fit, no it won’t; there’s too much liquid to fit in that drinking trough.
Okay the final question that we’re going to look at: we’re given a right regular
octagonal prism and we’re asked to find the surface area of this prism. So first of all, let’s just think about the different faces that we need to
Well the top and the base of this prism are both octagons, so we need to find the area
of those octagons, and then it also has eight rectangular faces around the edges, so we’re
looking for eight rectangles and two octagons.
So let’s start off by thinking about these octagons; they’re regular octagons. And
you need to recall from previous work that there is a formula for calculating the area of a
regular polygon, and is this formula here that the area of a regular 𝑛 sided polygon is equal to
𝑛𝑥 squared over four multiplied by cot of a hundred and eighty over 𝑛, where 𝑥 is
representing the side length of this polygon and cot remember is equal to one over
So for our octagon, the number of sides is eight, so we’re gonna have 𝑛 equals
eight, and the side length is three, so we’re going to substitute 𝑥 is equal to three. So the area of each of these rectangles is gonna be eight multiplied by three
squared all over four multiplied by cot of one hundred and eighty over eight.
That will give me
one of these octagons, but remember they are the same on the top and the base. Therefore, I need
to double this in order to get the total area of the two. Now if I evaluate this using my calculator, I get an answer of eighty-six point
nine one one six for the area of those two together.
Now remember I’ve expressed the angle in
terms of degrees here because it’s one hundred and eighty, so I need to make sure my
calculator is in degree mode when I’m typing this in. Okay next we want to work out the area of the rectangular faces, all the sides of
this octagonal prism.
So they are just rectangles and their dimensions are the three centimeters
which is the side length of the octagon and ten centimeters which is the height of the prism. So each of these faces will have an area of ten times three, but remember
there’re eight of them, so I need to multiply that by eight in order to work out the
So that gives me two hundred and forty as a contribution to the area from the
sides. So I’ve now got all the different parts of the area that I need, so to work out
the total surface area, I just need to add these together. So that value of eighty-six point nine one one six plus that value of two
hundred and forty, and so that gives me a final answer then of three hundred and twenty-six point
nine centimeters squared for the surface area of this prism.
Remember the question asked me to find it
to the nearest tenth, so that’s how I’ve rounded my answer.
So to summarize then, in this video, we’ve just looked at how to approach a couple
of more complex problems related to the surface area and the volume of prisms.