Explainer: Surface Areas of Cylinders

In this explainer, we will learn how to calculate surface areas of cylinders and solve word problems in real-world context.

In order to find the surface area of a cylinder we first need to consider its net. Let us look at the cylinder shown.

If you think about β€œunraveling ” the curved surface of the cylinder, this would become a rectangle with a width of β„Ž and a length that would be equal to the circumference of the circle: 2πœ‹π‘Ÿ. The top and bottom of the cylinder are both circles with radius π‘Ÿ. 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 πœ‹π‘Ÿ2 and the area of rectangle is calculated by multiplying its length and width: β„ŽΓ—2πœ‹π‘Ÿ=2πœ‹π‘Ÿβ„Ž. The total surface area is, therefore, 2Γ—πœ‹π‘Ÿ2+2πœ‹π‘Ÿβ„Ž, which simplifies to 2πœ‹π‘Ÿ2+2πœ‹π‘Ÿβ„Ž.

We can then use this formula to calculate the surface areas of cylinders. When 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.

Let us look at some examples.

Example 1: Finding the Surface Area of a Cylinder

Find, to the nearest tenth, the total surface area of a cylinder with a radius of 4 in and a height of 8 in.

Answer

In this question, we are told that π‘Ÿ=4 and β„Ž=8. We can substitute these values into the formula S.A.=2πœ‹π‘Ÿ2+2πœ‹π‘Ÿβ„Ž: S.A.=2πœ‹(4)2+2πœ‹(4)(8).

Simplifying, we get 32πœ‹+64πœ‹=96πœ‹.

The question, however, asks for the solution to the nearest tenth, so we need to evaluate and then round our answer: 96πœ‹=301.5928947….

We will use the calculator to evaluate 96πœ‹. So, our final answer is 301.6 square inches.

One thing to pay particular attention to is whether you are given the radius or the diameter of the cylinder. Remember that the diameter is double the radius so 𝑑=2π‘Ÿ. Let us look at an example of a question where we know the diameter.

Example 2: Finding the Surface Area of a Cylinder

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

Answer

Here, we have a cylinder with a diameter of 16 mm, so it has a radius of 8 mm. So, π‘Ÿ=8 and its height β„Ž=12. We then substitute these values into the formula S.A.=2πœ‹π‘Ÿ2+2πœ‹π‘Ÿβ„Ž: S.A.=2πœ‹(8)2+2πœ‹(8)(12).

Simplifying, we get 128πœ‹+192πœ‹=320πœ‹.

We have been asked to find the surface area to the nearest tenth, so we need to evaluate and round to find our answer: 320πœ‹=1,005.309649….

We will use the calculator to evaluate 320πœ‹. So, our final answer is 1,005.3 mm2.

You may be asked to find the lateral surface area of a cylinder. If this is the case 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 of the cylinder. Let us have a look at an example of this.

Example 3: 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 we need to find the area of just the curved surface of the cylinder.

The curved surface of the cylinder would β€œunravel” into a rectangle with a width which is the same as the height of the cylinder and a length which is the same as the circumference of the circle: 2πœ‹π‘Ÿ. So, the area of this rectangle, and, hence, the lateral surface area of the cylinder is 2πœ‹π‘Ÿβ„Ž. In this question, we have that the radius π‘Ÿ=13 and the height β„Ž=23. We can then substitute these values into the formula 2πœ‹π‘Ÿβ„Ž: 2πœ‹(13)(23).

Simplifying, we get 598πœ‹=1,878.672407….

The question asks us to round to the nearest tenth so our final answer is 1,878.7 square feet.

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

A further question that you could be asked is to find the surface area of a cylinder given its circumference. This type of question contains an additional step where we would need to work out the radius of the cylinder. Let us look at this in an example.

Example 4: Finding Surface Area of a Cylinder given Its Circumference

A cylinder has a circumference of 38πœ‹ in and a height of 33 in. What is the surface area of the cylinder in terms of πœ‹?

Answer

Firstly, it is worth noting that we have been asked to leave our answer in terms of πœ‹. This often means that you are expected to answer this question without a calculator.

Here, too, we have been given the circumference of the cylinder, so our first step is to work out the radius. We know that the circumference of a circle is calculated using the formula 2πœ‹π‘Ÿ. So, 2πœ‹π‘Ÿ=38πœ‹.

Dividing both sides by 2πœ‹ gives us π‘Ÿ=19in.

Remember that the surface area of a cylinder is calculated using the formula S.A.=2πœ‹π‘Ÿ2+2πœ‹π‘Ÿβ„Ž, so we need to substitute in the values of π‘Ÿ and β„Ž: 2πœ‹(19)2+2πœ‹(19)(33).

Simplifying, we get 722πœ‹+1,254πœ‹=1,976πœ‹.

As we are told to leave our answer in terms of πœ‹, we do not need to evaluate our answer; we just need to include the units: 1,976πœ‹ square inches.

Key Points

  • The formula for the surface area of a cylinder is 2πœ‹π‘Ÿ2+2πœ‹π‘Ÿβ„Ž.
  • The formular for the area of just the lateral surface (curved surface) is 2πœ‹π‘Ÿβ„Ž.
  • It is often a good idea to complete all of your working out in terms of πœ‹, and only evaluate your answer at the end if you are asked to do so.

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