Math 9

Unit 1 Square Roots and Surface Area

Lesson 914 Surface Areas of Other Composite Objects

Getting Started

There are a few things to cover BEFORE we actually get into the lesson.

In the previous lesson we talked about important vocabulary, the idea of a RIGHT RECTANGULAR PRISMS

In this lesson, we will create composite shapes that include other shapes, including right circular cylinders

As always,  KNOWING what those words mean, having a visual picture, will be enormously helpful to your understanding.

You saw this in the previous lesson, an interactive website on right prisms,  LINK

Video Tutorial (right prisms)

Another interactive website, for some information on cylinders.   LINK

Video Tutorial (cylinders introduction)

The surface area of a right circular cylinder

Optional

ALL of this section is optional.
The textbook has a SHORT explanation of how to find the surface area of a cylinder.
If you are satisfied with that,  just skip this whole section!
However...  I can tell you that I have spent many, many, many hours explaining the formula to students, and so I have included a DETAILED explanation in the hopes that you will be able to move forward from here with a SOLID understanding of this formula.

Motivation

The formula for finding the surface area of a right circular cylinder  is not complicated,  and the textbook explains it at the top of page 36.

However...  when I work with students, they are often confused by the formula, how to use it, etc.

KNOWING WHY the formula is what it is,  is the best way to move forward.

I did this two ways.

Taking some pictures of a familiar cylinder (a toilet paper tube) and deriving the formula.
(you can go through this STEP BY STEP and ask questions about specific steps, and focus on each part as you read it.)

I also created a video of the process.

The surface area of a right circular cylinder  (developed in pictures)

Toilet paper roll

The inside of a toilet paper roll is a good model for a right circular cylinder.
Note that it has no ends!
In calculations involving cylinders, you need to consider whether the ends are included or not.

Important measurements (vocabulary)

Consider the cylinder resting on a circular base.
The height of the cylinder is the measurement shown.
The radius of the cylinder is the radius of the circular ends.
The cylinder has a radius measurement even if the circular ends aren’t there.

Cut the cylinder

This is the IMPORTANT IDEA.

If we open the cylinder, we end up with a familiar shape.

Cut the cylinder vertically.....

When you open it up,  you get a rectangle....  (if you cut it “perfectly”)

The dimensions of the rectangle

The key question is,  what are the dimensions of the rectangle?   
How does the rectangle relate to the original cylinder?
DO your best to make the connection before you look at the next section....

Cylinder ↔ Rectangle

Take a cylinder,  imagine “opening it up”.

You can easily do this for yourself with a sheet of writing paper.... roll it up, unroll it,  and connect the dimensions of the rectangle with parts of the cylinder...

The height of the cylinder is one of the dimensions of the resulting rectangle.... and the other dimension of the rectangle would be the circumference of the circular end of the cylinder

Summary

The circumference of the circle that was at the top of the tube has become the length of the rectangle
The height of the tube has become the width of the rectangle

Since the formula for the area of a rectangle is A = l × w,   we can easily calculate the surface area of the tube, by using the formula for the area of a rectangle.

It looks complicated,  but  A = l ×w  becomes  A = (2 π r) h

The total surface area

If all that worked for you, use this interactive website and (hopefully)  the formula will make sense!!

LINK   (deriving the surface area of a right cylinder)

Surface area of a right circular cylinder:

A circle at each end.

The area of a circle is  A = π r

Two circles,  so   2 π r

When we “unroll” the cylinder part,  we get a rectangle.

The dimensions of the rectangle are the height of the cylinder, and the circumference of the circular end.

So, the area of the rectangle is  A = (2 π r) h

Put those together,     S.A(Subscript[., cylinder])= 2 π (r^2)+ 2 π r h

Video development

I did a whole series of videos trying to develop this idea one small step at a time.
Here are the important ones.

Formula A

Formula B

Formula C

Two examples

Below,  links to two video examples where I find the surface area of a right circular cylinder

Example 1

Example 2

Focus

The FOCUS section introduces the main concept that will be covered in the lesson.

TEXT Page 33

Read/Work through the  Focus  section at the top of page 33 in  your textbook.

Watch

Video Tutorial

Investigate

Investigate an idea or problem, with a partner if possible.
The investigation provides the background for the ideas presented in the lesson.

TEXT Page 33

Read/Work through the  Investigate  section on Page 33 in your textbook.

Watch

Video Tutorial

Connect

Connect summarizes the math in the lesson.
A presentation of ideas connected to the lesson focus.

TEXT Pages 33-34, 36

Read/Work through the  Connect  section on Pages 33 - 34, 36  in your textbook.
(there is a section of notes, then two examples, then another section of notes on page 36)

I didn’t create a video of this,  since the content will be covered over and over in all the other notes and examples and practice problems for this lesson.
Read through the material, do your best to follow, and then develop your understanding by working through all the other parts of this lesson.

Special overlap technique

An explanation of a special technique that can make dealing with the overlap in composite objects MUCH SIMPLER.
You certainly don’t need to do things this way, but because problems involving composite shapes can be quite complicated,  I think it’s worth doing anything you can to help make it simpler!

Overlap technique video tutorial

Examples

Examples will give you TOOLS for solving problems, and also show you that there may be different ways to approach a question.

Example 1  Determining the Surface Area of a Composite Object Made from Two Prisms

TEXT Pages 34 - 35

Work through Example 1 on Pages 34 - 35  in your textbook.

Watch

Video Tutorial A  (introduction)

Video Tutorial B (consider two separate pieces, then subtract overlap)

Video Tutorial C (work with the actual object and add areas of faces)

Example 2  Determining the Surface Area of a Composite Object Made from Two Cylinders

TEXT Pages 36 - 37

Work through Example 2 on Pages 36 - 37  in your textbook.

Watch

Video Tutorial A  (introduction)

Video Tutorial B  (calculations)

Example 3  Using the Pythagorean Theorem in Surface Area Calculations

TEXT Pages 38 - 39

Work through Example 3 on Pages 38 - 39  in your textbook.

Watch

Video Tutorial A  (introduction)

Video Tutorial B  (calculations, done like the book)

Video Tutorial C  (short... perhaps a slightly better way to get the answer....?)

Practice Problems

Doing Practice Problems

Practice problems reinforce the math presented in this lesson.

•	Work through the practice problems for this lesson.
	These problems are for practice, you don’t need to hand them in!
	(your teacher may assign specific questions for you to complete)

•	Use the answers in the back of the book to CHECK YOUR WORK!

•	If you make a mistake, TRY TO FIX IT!

•	If you need help or want more information, view the video solutions provided.

•	If you still need more help,  CONTACT YOUR TEACHER!

Types of Practice Problems

Discuss the Ideas invites you to talk about the math.
If possible share your thoughts/ideas/answers with another person.
If you can help them understand, then you likely have good understanding yourself.

Check questions require basic understanding of the ideas of the  lesson.

Apply questions will require you to dig deeper for a solution.

Take It Further questions offer enrichment and extension.

Notes

General note

NOTE!   I really want to emphasize that there are often many different ways to think about and tackle these problems.
I don’t want to make things complicated and present multiple solutions for each question.
These video solutions are just ONE POSSIBLE METHOD and not necessarily the best way!
I am trying to be clear and helpful and that usually means NOT taking shortcuts or trying to be super efficient with every step.

Question 7

In the second part of this question they ask you to come up with reasonable dimensions for a door and two windows.
You should do that!!
But...  if you want to check your work,  perhaps watch the video and once you have seen my dimensions, work with those.
(but make sure yours are close!)

Question 12

Sorry,  I tried here, but can’t really make sense of this question.   The canvas needed for the tent trailer would have to have some “slack” to be able to put that bar in there at the end...?
Can’t really understand exactly how that would work...  if you have some ideas let me know!

Question 16

In the interests of time,  and thinking not too many students will design their own house model....!,   I skipped this one.
Great question, but trying to use my time wisely.
Would love to see any student designs and would check your work if you do this....

Video Solution Links

Check your work for every problem you do!
If you make a mistake,  try to fix it!
Use the video solutions for more information.
If a problem still doesn’t make sense, ASK YOUR TEACHER FOR HELP!

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