Video: Surface Areas of Cylinders

Learn how to calculate the lateral and total surface area of a cylinder from its height and radius or diameter. Apply this knowledge to a problem that involves working backward from a known surface area and radius to calculate a missing height.

11:27

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.

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