Lesson Explainer: Surface Areas of Cylinders | Nagwa Lesson Explainer: Surface Areas of Cylinders | Nagwa

Lesson Explainer: Surface Areas of Cylinders Mathematics • 8th Grade

In this explainer, we will learn how to calculate surface areas of cylinders and use them to solve problems in real-life situations.

In order to find the surface area of a cylinder, we first need to consider its net. Let’s look at the cylinder shown, which has radius 𝑟 and height .

If you think about “unraveling” the curved surface of the cylinder, this would become a rectangle with a width of and a length that would be equal to the circumference of the circle: 2𝜋𝑟. The top and bottom of the cylinder are both circles with radius 𝑟. The net of the cylinder can, therefore, be drawn out as follows.

To find the surface area of the cylinder, we need to work out the area of its net. The area of each of the circles is 𝜋𝑟, and the area of the rectangle is calculated by multiplying its length and width to get ×2𝜋𝑟=2𝜋𝑟. Therefore, the total surface area is 2×𝜋𝑟+2𝜋𝑟, which simplifies to 2𝜋𝑟+2𝜋𝑟.

While you are gaining confidence with the method, it is recommended that you draw out the cylinder’s net to help you remember the method and how to derive the formula.

Note that some questions on this topic use the term right circular cylinder. This is just another way to describe an “ordinary” cylinder, which has its top circular face parallel to, and directly above, the bottom circular face.

Let’s look at an example featuring the net of a cylinder.

Example 1: Calculating the Surface Area of a Cylinder Formed by Folding a Rectangle

The diagram below shows the net of a cylinder where 𝐴𝐵𝐶𝐷 is a rectangle with 𝐴𝐵=20cm and 𝐴𝐷=44cm. The net is formed into a cylinder by joining 𝐴𝐵 with 𝐷𝐶, then folding over the two circles of radii 7 cm to make the top and the base.

What is the total surface area of the cylinder? Use 𝜋 as 227.

Answer

Recall that the total surface area of a cylinder is equivalent to the area of its net and that for a cylinder of radius 𝑟 and height , we have the formula totalsurfaceareaareaofcirclesareaofrectangleareaofonecirclewidthheight=+=(2×)+(×)=2×𝜋𝑟+(2𝜋𝑟×).

By using information from the question, we can substitute into this formula to find the total surface area of the cylinder.

First, we have that the circles forming the top and base have radii of 7 cm, so 𝑟=7.

Then, from the diagram, rectangle 𝐴𝐵𝐶𝐷 has a height of 20 cm and a width of 44 cm. The height of the rectangle will be the same as the height of the cylinder, so =20. Similarly, the rectangle’s width is the same as the circumference of a circular face (which has a length of 2𝜋𝑟), so 2𝜋𝑟=44.

Therefore, the total surface area of this cylinder is the sum of the areas of two circles of radius 𝑟=7 and a rectangle with width 2𝜋𝑟=44 and height =20; so, totalsurfacearea=2×𝜋𝑟+(2𝜋𝑟×)=2×𝜋×7+(44×20)=(2×𝜋×49)+880=(98×𝜋)+880.

Using the given approximation 227 for 𝜋, this becomes totalsurfacearea=98×227+880=987×22+880=(14×22)+880=308+880=1188.

All lengths were given in centimetres, so the total surface area must be in square centimetres. The total surface area of the cylinder is 1‎ ‎188 cm2.

As the total surface area of a cylinder is equivalent to the area of its net, then for a cylinder of radius 𝑟 and height , we have the following general formula.

Formula: Total Surface Area of a Cylinder

The total surface area of a cylinder of radius 𝑟 and height is given by the formula totalsurfacearea=2𝜋𝑟+2𝜋𝑟.

Since 𝜋=3.14159 is just a number, this means that as long as we know the radius and the height of a cylinder, we can always apply this formula to find its total surface area. Thus, in the first example, we could have substituted the values of and 𝑟 directly into this formula, which would have given the same answer.

In some questions, you may be asked to find the lateral surface area of a cylinder. If so, you only need to work out the surface area of the curved surface of the cylinder and not include the circles at the top and bottom. Therefore, we have a simpler formula for this case.

Formula: Lateral Surface Area of a Cylinder

The lateral surface area of a cylinder of radius 𝑟 and height is given by the formula lateralsurfacearea=2𝜋𝑟.

Again, as long as we know the radius and the height of a cylinder, we can always apply this formula to find its lateral surface area. Let’s look at an example of this type.

Example 2: Finding the Lateral Surface Area of a Cylinder

Determine, to the nearest tenth, the lateral surface area of the cylinder shown.

Answer

Here, we have been asked to find the lateral surface area of the cylinder, which means that we need to find the area of just the curved surface of the cylinder. Recall that the formula for the lateral surface area of a cylinder of radius 𝑟 and height is given by lateralsurfacearea=2𝜋𝑟.

The cylinder in this question has radius 𝑟=13 and height =23, so we can substitute these values into the formula to get lateralsurfacearea=2×𝜋×𝑟×=2×𝜋×13×23=𝜋×598=1878.672.

We were asked to round our answer to the nearest tenth. Remember that the tenths digit is the first digit after the decimal point, which in this case is a 6. The digit following this (the hundredths digit) is a 7, so the answer rounds up to 1‎ ‎878.7 to the nearest tenth.

Since the radius and height of the cylinder were given in feet, the lateral surface area must be in square feet. The lateral surface area of the cylinder, rounded to the nearest tenth, is 1‎ ‎878.7 ft2.

You may have noticed that in the examples above, we showed all of our working in terms of 𝜋 and only evaluated the answers at the end of the working. This is an excellent habit to get into as it will help to avoid any rounding errors and will improve your algebraic skills. Also, some questions will ask you to leave your answer in terms of 𝜋, so you need to be confident doing this anyway.

Next, we will work through an example where we are given the diameter of the cylinder, not the radius. Our approach is very similar but with one additional step. Always make sure to check whether you are given a radius or a diameter in the question.

Example 3: Finding the Surface Area of a Cylinder

Find, to the nearest tenth, the total surface area of this cylinder.

Answer

Recall that the total surface area of a cylinder of radius 𝑟 and height is given by the formula totalsurfacearea=2𝜋𝑟+2𝜋𝑟.

Notice that the diagram above shows a cylinder with a height of 12 mm and a diameter of 16 mm. To substitute into the formula, we need to know the radius 𝑟, which is half of the diameter. Therefore, our first step is to calculate the radius by dividing the diameter by 2, which gives 𝑟=16÷2=8. We can then substitute this value into the formula together with =12 to get totalsurfacearea=2×𝜋×𝑟+(2×𝜋×𝑟×)=2×𝜋×8+(2×𝜋×8×12)=(2×𝜋×64)+(2×𝜋×96)=128𝜋+192𝜋=320𝜋=1005.309.

From the question, we need to round our answer to the nearest tenth. The tenths digit is the first digit after the decimal point, which here is a 3. The digit following this (the hundredths digit) is a 0, so our answer must round down to 1‎ ‎005.3 to the nearest tenth.

The diameter and height of the cylinder were given in millimetres, so the volume is in square millimetres. We conclude that the total surface area of the cylinder is 1‎ ‎005.3 mm2, rounded to the nearest tenth of a square millimetre.

Note that formulas for the lateral and total surface areas are expressed in terms of the variables 𝑟 and . The lateral surface area formula is the simpler of the two, and we can always work backward from it to calculate the radius or height of a cylinder if given the lateral surface area and one of the other two measurements. The total surface area formula is more complicated, but we can always work backward from it to calculate the height of a cylinder if given the total surface area and the radius.

The next example shows how to rearrange the lateral surface area formula to solve this type of problem. In general, if a question refers to “the surface area,” we may assume that this means the total surface area. Otherwise, if a question involves the lateral surface area, we will always be told this explicitly, as is the case here.

Example 4: Finding the Diameter of a Cylinder given Its Lateral Surface Area and Height

Find the diameter of the base of a cylinder if the lateral surface area is 434𝜋 square centimetres and the height is 31 centimetres.

Answer

Recall the formula for the lateral surface area of a cylinder of radius 𝑟 and height : lateralsurfacearea=2𝜋𝑟.

In this question, we are given the lateral surface area and the height of the cylinder, and we are asked to find the diameter. Therefore, our method will be to substitute into the formula and solve the resultant equation for the radius 𝑟. Once we have done this, we can double our answer to get the diameter.

Taking the lateral surface area formula and substituting the values 434𝜋 for the lateral surface area and 31 for , we get 434𝜋=2×𝜋×𝑟×31=62𝜋𝑟.

Dividing both sides by 62 and then 𝜋 gives 7=𝑟.

Thus, we have calculated the radius of the cylinder, but since we were asked to find the diameter, we must double this value to get 2×7=14.

As the height was given in centimetres, then the diameter will have the same unit of measurement; the diameter of the cylinder is 14 cm.

In our final example, we are given the total surface area of the cylinder together with a relationship between the height and radius. This time, we will need to work backward from the total surface area formula to find the height of the cylinder.

Example 5: Finding the Height of a Cylinder given its Radius and Surface Area

A cylinder has a surface area of 72 cm2 and its height is equal to its base radius. Find the height of the cylinder, giving your answer to two decimal places.

Answer

Recall that the total surface area of a cylinder of radius 𝑟 and height can be calculated by using the formula totalsurfacearea=2𝜋𝑟+2𝜋𝑟.

Here, we are given the surface area of the cylinder and a relationship between the height and radius. Our strategy will be to substitute this information into the formula and rearrange to find the height.

Since the surface area is 72𝜋, then in the formula, we have 72𝜋=2𝜋𝑟+2𝜋𝑟.

Also, the height is equal to the base radius, so =𝑟. Since we are asked to find the height, then must appear in our equation, so we replace 𝑟 with throughout to get 72𝜋=2𝜋+2𝜋=4𝜋.

Dividing both sides by 4 and then 𝜋 gives 18=.

Finally, we take the square root of both sides, so 18=, which simplifies to 4.242=.

This is the same as =4.242, and this value rounds to 4.24 to two decimal places.

As the surface area was given in square centimetres, then the height must be in centimetres. We have found that the height of the cylinder, rounded to two decimal places, is 4.24 cm.

Let’s finish by recapping some key concepts from this explainer.

Key Points

  • The total surface area of a cylinder of radius 𝑟 and height is given by the formula totalsurfacearea=2𝜋𝑟+2𝜋𝑟.
  • The lateral surface area of a cylinder of radius 𝑟 and height is given by the formula lateralsurfacearea=2𝜋𝑟.
  • Always make sure to check whether you are given the radius or the diameter of the cylinder in the question.
  • For reasons of accuracy, it is often a sensible idea to complete all of your working in terms of 𝜋 and only evaluate the answer at the end if you are asked to do so.
  • We can always work backward from the lateral surface area formula to calculate the radius or height of a cylinder if given the lateral surface area and one of the other two measurements.
  • We can always work backward from the total surface area formula to calculate the height of a cylinder if given the total surface area and the radius.

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