Video Transcript
In this video, we are going to look
at how to calculate the surface area of a cylinder and then apply this method to a
couple of questions. So let’s think about calculating
the surface area of a cylinder. Now you would have seen before
that, in order to calculate the surface area of a prism, you need to find the total
area of all of its faces. So what we need to think about are
what are the different faces that the cylinder has. Now one way to do that is to sketch
the net of this cylinder, so what it would look like in two dimensions if we were
able to unwrap it.
So first of all, the cylinder has
two circular faces. It has the faces that are on the
top and the base of the cylinder, the ones that I’m shading in here. So when we draw the net of this
cylinder, we need to include these two circular faces. Now I’ve labelled my diagram. It has the measurements ℎ and 𝑟 on
it; ℎ is standing for the height of the cylinder and 𝑟 is standing for the radius
of the circle. So there are those two circular
faces. Now a cylinder also has a curved
surface, which is this one that I’m currently shading in in orange here. And think about what the shape of
that would be.
Even though it’s a curved surface,
if you were to unfold it so that it was a flat surface, you would see that it’s
actually a rectangle. And you could try this
yourself. If you find a food can or something
like that and unpeel the label, you will see that the shape of that label is in fact
a rectangle. So if I add it to my net, I’ll see
that the net of the cylinder looks something like this, a rectangular part and then
these two circular parts. So now I need to work out what the
area of each of these parts are. Well the circle, remember the area
of a circle, the formula for that is 𝜋𝑟 squared, so each of those circles will be
contributing that amount to the area. So therefore, I will have a total
contribution of two 𝜋𝑟 squared from the circular top and the circular base.
Now let’s think about calculating
the area of this rectangle. So for rectangle, we do the length
multiplied by the width while this measurement here, that’s just the height of the
cylinder. So that’s ℎ. And then I need to think about what
is the width of the rectangle of this measurement here. Well, if you think about this
carefully, or perhaps if you think back to your food can and unwrapping the label,
you’ll see that this measurement here matches up perfectly with the circumference of
the circle on the top and on the base. And if you recall then, the formula
for calculating the circumference of a circle is either 𝜋𝑑 where 𝑑 represents the
diameter or two 𝜋𝑟 where 𝑟 represents the radius. And as it’s the radius that I’ve
used in other parts of this calculation, it’s the radius that I will again use
here. So the dimensions of the rectangle
then are ℎ and two 𝜋𝑟, which means the area of this rectangle when I multiply them
together will be two 𝜋𝑟ℎ.
So now I have all that I need in
order to write down the formula for calculating the total surface area of the
cylinder. It’s just gonna be the sum of these
three areas. So here is that formula. The total surface area of the
cylinder is equal to two 𝜋𝑟 squared plus two 𝜋𝑟ℎ. Now when you’re working with
cylinders, sometimes you aren’t asked to calculate the total surface area. You’re asked instead to calculate
something called the lateral surface area. And what this refers to is the
curved part of the surface area. So in our net of the cylinder, it’s
this rectangular part here. So if you’re just asked to
calculate the lateral surface area, then you’ll be using this formula instead, which
is that it’s equal to two 𝜋𝑟ℎ as it’s only the area of that rectangle that we’re
interested in. So let’s look at applying this to a
question.
The question asked us to calculate
the lateral surface area of the cylinder below. And we can see this cylinder has a
height of 13 centimetres and a diameter of 10 centimetres. So first of all, note the question
has asked for the lateral surface area. So we’re only interested in that
curved part, not the full surface area including the top and the base. So remember the formula that we
need here is that the lateral surface area is equal to two 𝜋𝑟ℎ. So in this question, ℎ, well that’s
13 centimetres, and 𝑟 remember presents the radius. We haven’t got the radius. We’ve got the diameter which is
10. So the radius is half of that which
is five. So our calculation then is the
lateral surface area is two times 𝜋 times five times 13.
Now actually remember that that two
𝜋𝑟 in the formula for the lateral surface area is representing the circumference
of the circle. And another way to write that would
have been 𝜋𝑑. So rather than dividing this value
of 10 by two to get the radius and including the factor of two in the formula, we
could actually have just done 𝜋𝑑ℎ and then 𝜋 times 10 times 13. So rather than dividing by two and
multiplying by two, we could have got about in a slightly simpler way.
At any way, it gives us the same
result in our calculation. So two times 𝜋 times five times
13, this gives me 130𝜋 as the lateral surface area. I could leave my answer in terms of
𝜋, if that’s how it was requested or if I didn’t have a calculator. But if I go on and evaluate my
answer as a decimal, then it gives me an answer of 408.4 centimetres squared. And that’s been rounded to one
decimal place or the nearest tenth. Units, remember, we’re talking
about area. So I need to square units with my
answer, which is why the centimetres squared in this instance.
Right, let’s look at our next
question on this.
This time we are asked to calculate
the total surface area of the cylinder below.
So I need to recall the formula for
calculating the total surface area. Remember, that is going to include
the top and the base this time. So here is the formula we need. The total surface area is equal to
two 𝜋𝑟 squared plus two 𝜋𝑟ℎ. So it’s just a question of
substituting the correct values into this formula. The radius is six millimetres and
the height is 25 millimetres. So the calculation will be two
times 𝜋 times six squared plus two times 𝜋 times six times 25. The brackets, I don’t actually need
them; they’re not mathematically necessary. I’ve just put them in so that you
can see the two parts of the calculation more clearly.
So a couple of things to watch out
for then, have you been asked for the total surface area or the lateral surface
area? So do you need to include the top
and the base or not. And also have you been given the
radius or have you been given the diameter of the cylinder? And if you’ve been given the
diameter, you just need to think carefully about how you’re going to use it in the
formula. So if I evaluate these, then I have
72𝜋 plus 300𝜋. So that gives me an overall total
of 372𝜋 for the total surface area. Again, I could leave my answer at
this stage, but I’ll evaluate it as a decimal. And doing that gives me an answer
of 1168.7 millimetres squared. And again, that answer has been
rounded to one decimal place.
Okay, this is the final question
that we’re going to look at in this video.
It says the total surface area of a
cylinder of radius two meters is 28𝜋 meters squared. Calculate the height of the
cylinder.
So this question is an example of
one when we need to work backwards. We’re given the surface area and we
have to work out one of the missing dimensions of the cylinder. So first of all, we need to recall
the relevant formula. And reading the question carefully,
it says the total surface area. So we need the formula for total
rather than lateral. So it’s this formula here, total
surface area is equal to two 𝜋𝑟 squared plus two 𝜋𝑟ℎ. So we can use this formula to form
an equation involving the radius and the total surface area that we do know and the
height that we don’t. Once we’ve got the equation, we’ll
will be able to solve it in order to work out what this height is.
So let’s form this equation. I’m gonna substitute 𝑟 was equal
to two into this formula. So two 𝜋𝑟 squared is gonna be two
times 𝜋 times two squared. And then two 𝜋𝑟ℎ is gonna be two
times 𝜋 times two times this value ℎ that we don’t know. And we’re told that is all equal to
28𝜋. So I’ve set up the start of my
equation. Now I can simplify it a bit. So looking at the first term,
that’s gonna simplify. Well two squared is four and then
multiplied by two is eight. So I have eight 𝜋 for that first
term. And then the second term, two times
two is four, so I’m gonna have four 𝜋ℎ for that second term. So that brings me to this stage
here. Eight 𝜋 plus four 𝜋ℎ is equal to
28𝜋. Now all of those terms have a
factor of 𝜋 in, so actually I can divide through by 𝜋 or cancel out that factor to
simplify this equation a little bit.
If I just write it out again then
so we can see a little bit more clearly, so I can see that my equation is actually
just four ℎ plus eight is equal to 28. Solving this equation that is
relatively straightforward, the first step is I need to subtract eight from both
sides of the equation. And so that gives me four ℎ is
equal to 20. And then I need to divide both
sides of the equation by four. And so that gives me the ℎ is equal
to five. ℎ, remember, represents the height
so it needs a unit. And if you look back at the
question, all of the units were meters and meters squared, so this must be five
meters for the height of the cylinder. So this question involved using the
information that we were given to set up and then solve an equation in order to work
out a missing dimension of the cylinder.
To answer a similar question where
it was the radius that was missing would be a more complex problem, as it would
result in a quadratic equation to solve. But that is a possible extension of
this. So to summarise then, we’ve looked
at how to calculate the lateral and the total surface area of a cylinder. We’ve seen where that formula comes
from in terms of the net of the cylinder. We’ve applied it to a couple of
more straightforward questions where we just have to substitute the relevant
values. And then finally, we’ve looked at a
question that involves working backwards from a known surface area to calculate the
height of the cylinder.
