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Video: Surface Areas of Cylinders

Lauren McNaughten

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.

11:28

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 letโ€™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 Iโ€™m shading in here. So when we draw the net of this cylinder, we need to include these two circular faces. Now Iโ€™ve labelled my diagram. It has the measurements โ„Ž and ๐‘Ÿ on it; โ„Ž is standing for the height of the cylinder and ๐‘Ÿ 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 Iโ€™m currently shading in in orange here. And think about what the shape of that would be. Even though itโ€™s a curved surface, if you were to unfold it so that it was a flat surface, you would see that itโ€™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, Iโ€™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 ๐œ‹๐‘Ÿ squared, so each of those circles will be contributing that amount to the area.

So therefore, I will have a total contribution of two ๐œ‹๐‘Ÿ squared from the circular top and the circular base. Now letโ€™s think about calculating the area of this rectangle. So if a rectangle, we do the length multiplied by the width while this measurement here, thatโ€™s just the height of the cylinder. So thatโ€™s โ„Ž. And then I need to think about what is the width of the rectangle this measurement here. Well if you think about this carefully or perhaps if you think back to your food can and unwrapping the label, youโ€™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 are cool, then the formula for calculating the circumference of a circle is either ๐œ‹๐‘‘ where ๐‘‘ represents the diameter or two ๐œ‹๐‘Ÿ where ๐‘Ÿ represents the radius. And as itโ€™s the radius that Iโ€™ve used in other parts of this calculation, itโ€™s the radius that I will again use here.

So the dimensions of the rectangle then are โ„Ž and two ๐œ‹๐‘Ÿ, which means the area of this rectangle when I multiply them together will be two ๐œ‹๐‘Ÿโ„Ž. So now I have all that I need in order to write down the formula for calculating the total surface area of the cylinder. Itโ€™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 ๐œ‹๐‘Ÿ squared plus two ๐œ‹๐‘Ÿโ„Ž. Now when youโ€™re working with cylinders, sometimes you arenโ€™t asked to calculate the total surface area. Youโ€™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 a cylinder, itโ€™s this rectangular part here. So if youโ€™re just asked to calculate the lateral surface area, then youโ€™ll be using this formula instead, which is that itโ€™s equal to two ๐œ‹๐‘Ÿโ„Ž as itโ€™s only the area of that rectangle that weโ€™re interested in.

So letโ€™s look at a prim 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 thirteen centimetres and a diameter of ten centimetres. So first of all, note the question has asked for the lateral surface area. So weโ€™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 ๐œ‹๐‘Ÿโ„Ž. So in this question, โ„Ž, well thatโ€™s thirteen centimetres, and ๐‘Ÿ remember presents the radius. We havenโ€™t got the radius. Weโ€™ve got the diameter which is ten. So the radius is half of that which is five. So our calculation then is the lateral surface area is two times ๐œ‹ times five times thirteen.

Now actually remember that that two ๐œ‹๐‘Ÿ in the formula for the lateral surface area is representing the circumference of the circle. And another way to write that would have been ๐œ‹๐‘‘. So rather than dividing this value of ten by two to get the radius and including the factor of two in the formula, we could actually have just done ๐œ‹๐‘‘โ„Ž and then ๐œ‹ times ten times thirteen. 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 ๐œ‹ times five times thirteen, this gives me a hundred and thirty ๐œ‹ as the lateral surface area. I could leave my answer in terms of ๐œ‹, if thatโ€™s how it was requested or if I didnโ€™t have a calculator. But if I go on and evaluate my answer as a decimal, then it gives me an answer of four hundred and eight point four centimetres squared. And thatโ€™s been rounded to one decimal place or the nearest tenth. Units, remember, weโ€™re talking about area. So I need to square units with my answer, which is why the centimetres squared in this instance.

Right, letโ€™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 ๐œ‹๐‘Ÿ squared plus two ๐œ‹๐‘Ÿโ„Ž. So itโ€™s just a question of substituting the correct values into this formula. The radius is six millimetres and the height is twenty-five millimetres.

So the calculation will be two times ๐œ‹ times six squared plus two times ๐œ‹ times six times twenty-five. The brackets, I donโ€™t actually need them. Theyโ€™re not mathematically necessary. Iโ€™ve just put them in so that you can see the two parts of the calculation more clearly. So if I evaluate these, then I have seventy-two ๐œ‹ plus three hundred ๐œ‹. So that gives me an overall total of three hundred and seventy-two ๐œ‹ for the total surface area. Again, I could leave my answer at this stage, but Iโ€™ll evaluate it as a decimal. And doing that gives me an answer of one thousand one hundred and sixty-eight point seven millimetre squared. And again, that answer has been rounded to one decimal place.

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 youโ€™ve been given the diameter, you just need to think carefully about how youโ€™re going to use it in the formula.

Okay this is the final question that weโ€™re going to look at in this video. It says the total surface area of a cylinder of radius two meters is twenty-eight ๐œ‹ meters squared. Calculate the height of the cylinder. So this question is an example of one when we need to work backwards. Weโ€™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 itโ€™s this formula here, total surface area is equal to two ๐œ‹๐‘Ÿ squared plus two ๐œ‹๐‘Ÿโ„Ž. 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 donโ€™t. Once weโ€™ve got the equation, weโ€™ll will be able to solve it in order to work out what this height is.

So letโ€™s form this equation. Iโ€™m gonna substitute ๐‘Ÿ was equal to two into this formula. So two ๐œ‹๐‘Ÿ squared is gonna be two times ๐œ‹ times two squared. And then two ๐œ‹๐‘Ÿโ„Ž is gonna be two times ๐œ‹ times two times this value โ„Ž that we donโ€™t know. And weโ€™re told that is all equal to twenty-eight ๐œ‹. So Iโ€™ve set up the start of my equation. Now I can simplify it a bit. So looking at the first term, thatโ€™s gonna simplify. Well two squared is four and then multiplied by two is eight. So I have eight ๐œ‹ for that first term. And then the second term, two times two is four, so Iโ€™m gonna have four ๐œ‹โ„Ž for that second term. So that brings me to this stage here. Eight ๐œ‹ plus four ๐œ‹โ„Ž is equal to twenty-eight ๐œ‹. Now all of those terms have a factor of ๐œ‹ in, so actually I can divide through by ๐œ‹ 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 โ„Ž plus eight is equal to twenty-eight.

Solving this equation, it is relatively straightforward. The first step is I need to subtract eight from both sides of the equation. And so that gives me four โ„Ž is equal to twenty and then I need to divide both sides of the equation by four. And so that gives me the โ„Ž is equal to five. โ„Ž, 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, weโ€™ve looked at how to calculate the lateral and the total surface area of a cylinder. Weโ€™ve seen where that formula comes from in terms of the net of the cylinder. Weโ€™ve applied it to a couple of more straightforward questions where we just have to substitute the relevant values. And then finally, weโ€™ve looked at a question that involves working backwards from a known surface area to calculate the height of the cylinder.