# Surface Area of a Prism and Cylinder

This article presents an intuitive approach to find the surface area of prisms and cylinders. Check out the SWT lesson Introduction to Surface Area if you would like a refresher on the definition.

## For Teachers and Parents

### What is this Hint System

The hint system is designed to allow students to engage with the problems with little to no support (Discovery/Inquiry-Based Math). The hints gradually give more and more information until we end up with a full explanation (Traditional Lecture). Since Math is not a ‘spectator sport,’ it is hoped that repeated use of this hint system in multiple lessons will encourage students to use fewer and fewer hints and learn to solve problems creatively for themselves. This is, of course, only one possible arrangement of these hints. An instructor will be better able to adapt the presentation of the hints to the progress a specific student has made. This is beneficial to the student until they learn to use the system and self-regulate. If the student needs hints, they can choose to set aside their particular approach temporarily while filling in the gaps in the order the text presents them. Later they can analyze how the approach they took compares to that of the text.

### A Smidge of Calculus

There is a subtle connection to the concept of limit, from calculus, within this lesson. Exposing students to these ideas early so they have horizon knowledge about what is coming down the track later is important. It gives their subconscious extra years to digest the ideas before they become everyday companions.

## Introduction

### Finding the Surface Area of a Prism: Your First Encounter

To start, I'd like to invite you to try to come up with a solution on your own. Take a few minutes and think about how surface area works. Which equations for the areas of 2-dimensional shapes might be useful? Can you create your own general algorithm? Please don't work too hard. If you find yourself getting frustrated, take a break or move on to the next section. There are some nice pointers to get you started. Please stop often and reassess if you have new ideas about how to solve the problem. The fewer hints you use, the stronger your creative and problem-solving skills will become. Remember there are lots of ways to solve these problems. If you invent one for yourself, which is not covered here, that is awesome!

## Finding the Surface Area of a Prism: A Visual Approach (part 1)

Now that you have tried to find the relationship, take a look at this app. Please leave the hint buttons (bottom left corner) alone for now. You can play with the full-screen button, action button, and dials as much as you like. You can also tip the figure by dragging it and zoom in and out (using the pinching gesture on mobile devices and the mouse scroll wheel otherwise).

What do you see? What does the dial do? What does the action button do?

Hint 1
There is a prism displayed when the app starts. What does the dial do?
Hint 2
The dial starts at 3. When it changes the prism changes. What is the correlation? Use the first (left most) hint button, once you have a guess (or need help making a guess).
Hint 3
When the dial is set to 3 the prism is a triangular prism. When the dial is set to 4 the prism is a square prism. When the dial is set to 5 the prism is a pentagonal prism. When the dial is set to 6 the prism is a hexagonal prism. Do you see a pattern?
Hint 4
It appears that the dial setting dictates the number of sides on the base of the prism (3 = tri, 5 = pent, 6 = hex, etc.). What does the action button do?
Hint 5
The action button seems to (un/re)fold the prism, by moving the faces. How does this help?
Hint 6
The goal is to find the surface area of the prisms. How does this unfolded, flattened shape help?
Hint 7
Areas are calculated for 2-dimensional shapes. The shape shown is now 2-dimensional and has the same area as the surface area of the original prism (the faces just got moved around). If you find the area of this shape you also find the surface area of the original prism. How might you do that?
Hint 8
The flattened 2-dimensional shape is a composite shape, made up of the faces of the prism. Does this help?
Hint 9
To find the area of the composite shape you can just add the areas of the simpler shapes it is made from (the faces of the prism). This is one way to calculate the surface area, but there is another. Can you see it?
Hint 10
There is no reason to calculate the base two separate times. You can save time by calculating it once and just doubling the answer. What about the rectangular faces?
Hint 11
In the examples given, all the rectangles are the same, so you could calculate one and then multiply by the number of sides. Can you think of another way to find this same value? Try to find a method that would be fast, but still works if the rectangular panels are all different areas.
Hint 12
All the rectangular faces together seem to form a new rectangle. Does that help?
Hint 13
Can you find the area of the big rectangle directly?
Hint 14
What are the dimensions of the big rectangle?
Hint 15
One dimension is the depth of the prism. What is the other?
Hint 16
Do not press the remaining hint button yet. Please proceed to the next section for more hints. You will be asked to return to this app in one of the hints in the next section.

## Finding the Surface Area of a Prism: A Visual Approach (part 2)

This app is very similar to the app in the previous section. Please play around with the dial and all the buttons, except the second (right most) hint button. What do you see? What is the same? What is different?

Hint 1
Much of the app is the same. The prism is oriented differently and the action button causes a new animation which (un/re)folds the prism differently (a star pattern). Everything else seems to be the same. How does this help you find the second dimension you are looking for (in hint 15 of part 1)?
Hint 2
Try the second hint button in this app. Does that help?
Hint 3
Try the second hint button in the app from part 1. Does that help?
Hint 4
Try using the action button a few times in the app from part 1 (now that the second hint is on). Does this help?
Hint 5
The second dimension of the big rectangle appears to be the same length as the perimeter of the base. You could find the perimeter and multiply it by the depth of the prism to find the area of all the side panels at once. This works even if the rectangular faces are all different (the depth must be the same because they come from a prism). This perimeter trick will be important in the cylinder section coming up next.
Hint 6
You now have the area of both bases (the multiply by two trick) and all the side panels (the perimeter trick). Add these numbers together to get the surface area. How might you represent this with an equation?
Hint 7
You could use the equation ${\displaystyle A_{prism-surface}=2A_{base}+Pd}$. This equation can be specialized for specific kinds of prisms, but all that algebra is a lot of effort, with very little payoff. It is best to either remember the relationship written above, or (even better) to re-derive it from the definitions of surface area and prisms, then specialize it to your particular case as you require.

## Finding the Surface Area of a Cylinder: Your First Encounter

Great! Now you know how to find the surface area of prisms. Next, now can you think of a way to extend your knowledge to find the surface area of a cylinder? Can you think of a way to leverage what you just learned about prisms? Give it a try. When you have a conjecture (or decide you have made a good enough effort) have a look at the next section.

## Finding the Surface Area of a Cylinder: A Visual Approach

Have a look at the app in this section. Please play around with the the dial and all buttons, except the second (right most) hint button. What do you see?

Hint 1
This app has a lot in common with the first app in this article. The left most hint button makes the prism see through. The dial changes the number of sides and the action button causes an animation that (un/re)folds the prism. What is different?
Hint 2
The dial changes in increments of 10 with a minimum value of 20 and a maximum value of 100. The animation caused by the action button has a little more flourish, but finishes in the same place as the first app did. Do you notice anything interesting when the number of sides gets large?
Hint 3
There is something to notice about the base of prism (or the entire 3-dimensional shape if you prefer). What is it?
Hint 4
What is the purpose of this section?
Hint 5
The purpose of this section is to find the surface area of a cylinder. As the number of side gets large the base looks more and more like a circle. Which means the overall shape begins to look more and more like a cylinder. How might you go about finding the surface area of such a shape?
Hint 6
The area of the bases can be found using the circle area equation, so that isn't too hard, but what can be done to find the area of the curved side?
Hint 7
There are no small rectangular faces on a true cylinder, but the large unrolled rectangle seems to still appear. You could try to find the area of the big rectangle. What area its dimensions?
Hint 8
One dimension is the depth of the cylinder, just like the prism. What is the other?
Hint 9
Try using the second hint button. Does that help?
Hint 10
What was the other dimension in the case of the prism?
Hint 11
In the prism's case, the other dimension was the perimeter. What would you use in this case?
Hint 12
You are looking for the perimeter of a circle. What is another word for that?
Hint 13
The circumference is another word for the perimeter of a circle. The second dimension of the large rectangle is the same length as the circumference of the circular base. How do you put it together to get the surface area of a cylinder?
Hint 14
You can multiply the depth by the circumference to get the area of the big rectangle, then add this to twice the area of the circular base. What would this look like as an equation?
Hint 15
The equation would look like ${\displaystyle A_{cylinder-surface}=2A_{base}+Cd}$. This is the same equation as for the prism, but with perimeter exchanged for the circumference. It is worth trying to make this equation more specific by filling in the equations for some of the variables. Can you do it?
Hint 16
Substituting in the circle area and circumference equations gives ${\displaystyle A_{cylinder-surface}=2\pi r^{2}+2\pi rd}$. This can be further simplified to speedup calculations. Can you see how?
Hint 17
You can common factor out ${\displaystyle 2\pi r}$ to arrive at ${\displaystyle A_{cylinder-surface}=2\pi r(r+d)}$. You can use this equation (or the one in hint 16) if you like. It is OK if you only remember the more general equation for the prisms too. You'll just have to remember that perimeter and circumference are basically the same, then use the appropriate equations from 2-dimensional geometry when solving for cylinders.

## Finding the Surface Area of a Prism and Cylinder: A Hands on Approach (Nets)

The image of a net of a triangular prism, in this section, links to a PDF containing 7 nets. The PDF is accessible if you click the image (then the details button, if you are using the mobile version of the website) and then click the big version of the image again on the next page. The idea is to cut along the dotted lines and fold along the solid ones, except for one solid line in the cylinder net. This special line indicates how much overlap to use to attach the curved side of the cylinder to itself. The rest are faces of the prisms and tabs to glue the shapes together (tape can be used too). You should see lots of similarities to the apps. There are many ways to layout these nets. They just need to fold up into the shape you desire. You will find nets for a triangular prism, cube, square prism, rectangular prism, pentagonal prism, hexagonal prism, and cylinder. With the exception of the rectangular prism, all the prisms have regular n-gons as bases. Prisms can of course be made with less symmetric bases, but only one such example is provided here. Hands on learners will benefit greatly from building these shapes. If you had trouble understanding what was going on in the apps, this is you chance to go through the hints again and compare to the nets. Sometimes holding and folding the shapes yourself can yield better geometric intuition.