SM-1
Pre-check
Circles and Prisms
Name _______________________________
1. Label the following parts of the circle. Then give reasonable values for the radius,
diameter and circumference in centimeters.
Radius
If the radius is 10 cm, then the
Diameter
diameter is _______ cm and the
Circumference
circumference is _______ cm.
2. Write the formula and/or describe in your own words the process you would use to
find the…
Perimeter of a rectangle
___________________________________________________
Area of a rectangle
___________________________________________________
Surface Area of a rectangular prism
___________________________________________________
Volume of a rectangular prism
___________________________________________________
Circumference of a circle
___________________________________________________
Area of a circle
___________________________________________________
Find the circumference and area for each circle below.
3.
radius = 4 in.
4.
C = _______
A = _______
C
Cylinder Savvy
diameter = 9 ft.
=
_______
A
=
_______
Student Materials Page 1 of 22
5. Sketch and label a circle with a radius of 1 in. Find its circumference and area.
Circumference = ________
Area = ________
Now double the radius to 2 in. Find the new circumference and area.
Circumference = ________
Area = ________
How was the circumference affected by the change in radius? _______________
How was the area affected?
________________________________________________________________
What would happen to the circumference and area if the original radius is 3 inches?
Why?
________________________________________________________________
Compare answers of students’ who used 3.14 and those who used the pi (π) key on
their calculators. How do their answers vary? _________ Which is more accurate?
_______________
Cylinder Savvy
Student Materials Page 2 of 22
6. In the space below, sketch and label a rectangular prism with dimensions 4cm, 5cm,
and 8cm. Find the surface area and volume of your prism.
Surface Area: _______
Volume: _______
Now, sketch the same prism, but double the side that is 4mm. Compute the new
surface are and volume of your prism.
Surface Area: _______
Volume: _______
How did the surface area change?
___________________________________________
How did the volume change?
_______________________________________________
What do you predict will happen to the surface area and volume if we now double the
5 millimeters to 10 millimeter?
________________________________________________________________
Predict how surface area and volume will change in the 2nd figure. Find surface areas
and volumes for both figures.
7.
2m
4m
7m
2m
Cylinder Savvy
7m
2m
Student Materials Page 3 of 22
SM-2
___________________
Name(s) ___________________
Surface Area of a Cylinder (Form A)
Goal: Become familiar with the formula for finding the surface area of cylinders.
To find the surface area of the cylinder, we will need to:
• Find the area of all surfaces (3 surfaces total)
• circle on top
• circle on bottom
• lateral surface (wrapped around top and bottom)
• add the 3 areas together
1. Label the following parts of this cylinder: radius (r=4cm), height (h=6cm), base,
and lateral surface. What shape is the base? ________________
2. Imagine you cut your cylinder into pieces and lay them out flat. Label each
radius and the height of the lateral surface (rectangular when laid flat).
3. To find the area of a circle, use the formula A = πr 2 .
What is the area of each of the circles in our cylinder? _________________
Put this information in the chart that follows on number 6.
4. We still need to find the area for the rectangle that wraps around the circles.
Because it wraps around the circles, its length is equal to the circumference of
each circle. Calculate the circumference of the circle using . C = 2πr
Circumference of the circle = ______________
5. The last step to finding the lateral surface area is to multiply its length by its
width. Do this substituting in the Circumference for length and inserting the
given height. The formula for this is 2πr ⋅ h .
Area of lateral surface = ______________________
Cylinder Savvy
Student Materials Page 4 of 22
6. Fill in the table to be sure you have all the necessary values.
Surface Name
Circle #1
Circle #2
Lateral Surface
Formula for Area
Total Surface Area:
2 ⋅ πr 2 + 2πr ⋅ h
Area
πr 2
πr 2
2πr ⋅ h
Once you have the area for each of the 3 surfaces, you can add them to find the
total surface area. Note the Total Surface Area formula above. Can you see
where the areas of the 2 circles are listed? Can you see where the area of the
lateral surface is listed?
Now that you know where the formula comes from, use it to find the surface area
of several cylinders. Rewrite the equation for each problem, substituting in the
correct values for the radius and height. Then simplify. The first one is done for
you.
SA = 2 ⋅ πr 2 + 2πr ⋅ h
7. r = 2in.
h = 10in.
8. r = 10ft.
h = 8ft.
9. d = 4m
h = 4m
11. d = 9yd
h = 5yd
12. r = 5in.
h = 20in.
SA = 2 ⋅ π (2) + 2π (2)(10)
= 2 ⋅ π (4 ) + 2π (20)
= 8π + 40π
= 48π ≈ 150.72 in.2
2
10. r = 3.5cm
h = 6cm
Cylinder Savvy
Student Materials Page 5 of 22
Name(s) ___________________
___________________
SM-3
Surface Area of a Cylinder with Manipulatives (Form B)
Goal: Become familiar with the formula for finding the surface area of cylinders.
Materials: cylindrical object (oatmeal container), scissors, ruler with centimeters
To find the surface area of the cylinder, we will need to:
• Find the area of all surfaces (3 surfaces total)
• circle on top
• circle on bottom
• lateral surface (wrapped around top and bottom)
• add the 3 areas together
1. Sketch, measure (in centimeters), and label the following parts of your
cylindrical object: radius, height, base, and lateral surface. What shape is the
base of your figure? ________
2. Cut your cylinder into three pieces and lay them out flat. Sketch and label all
necessary parts.
3. To find the area of one of the circles, use the formula A = πr 2 .
What is the area? _________________ Record your answer in the table that
follows. What do you notice about the other circle?
How will its area compare? _____________________________
Cylinder Savvy
Student Materials Page 6 of 22
4. Think carefully about the lateral surface of your cylinder.
What shape is it? ______________________
What do you notice about the length (or base) of the polygon?
________________________________________________________
You should be able to see how the lateral surface wrapped around each
circle, so its length is equal to each circle’s circumference. Measure with a
ruler and compare the following values.
Length of Lateral Surface using a ruler = _____________
Circumference of the circle using formula C = 2πr = ____________
How do they compare? __________________________________
Next, multiply the circumference by the height of the lateral surface to find the
area.
Area of lateral surface = ______________________
5. Fill in the table to be sure you have all the necessary values.
Surface Name
Circle #1
Circle #2
Lateral Surface
Formula for Area
Total Surface Area:
2 ⋅ πr 2 + 2πr ⋅ h
Area
πr
πr 2
2πr ⋅ h
2
Now that you know where the formula comes from, use it to find the surface area
of several cylinders. Rewrite the equation, substituting in your values for the
radius and height. Then simplify. The first one is done for you.
SA = 2 ⋅ πr 2 + 2πr ⋅ h
6. r = 2in.
h = 10in.
7. r = 10 t.
h = 8 ft.
8. d = 4 m
h=4m
SA = 2 ⋅ π (2) + 2π (2)(10)
= 2 ⋅ π (4) + 2π (20)
= 8π + 40π
= 48π 150.72 in.2
2
9. r = 3.5 cm
h = 6 cm
Cylinder Savvy
10. d = 9 yd
h = 5 yd
11. r = 5 in.
h = 20 in.
Student Materials Page 7 of 22
Name _______________
SM-4
Volume of a Cylinder
Finding the volume of a cylinder can be easier to remember if you use the same formula
for all prisms. For all right prisms, volume is simply:
Area of the Base x Height
Try it with these cylinders. Remember, the base of a cylinder is always a circle, so you
will always need to begin by finding the area of a circle.
1. r = 2 in.
h = 10 in.
2. r = 10 ft.
h = 8 ft.
3. d = 4 m
h=4m
5. d = 9 yd
h=5d
6. r = 5 in.
h = 20 in.
V=Area of Base · Height
= πr 2
·
h
2
= π (2)
· (10)
= π4 · (10)
= 40π ≈ 125.6 in.3
4. r = 3.5 cm
h = 6 cm
Cylinder Savvy
Student Materials Page 8 of 22
SM-5
Exploration: Paper & Rice
Goal: Demonstrate how changes in radius can affect the volume of cylinders without
affecting surface area.
Your group will need:
• 1 bag of rice
• 1 sheet of paper
• tape
• a ruler
• 1 box lid
• 1 pair of scissors
• 1 measuring cup
Follow these instructions:
1. Have one person in your group cut the piece of paper exactly in half (hamburgerstyle). Label the half sheets A and B.
2. Measure and find the surface area of rectangle A.
3. Measure and find the surface area of rectangle B.
4. Tape A into a hotdog-style cylinder (longways).
5. Tape B into a hamburger-style cylinder (wider).
6. Predict below which cylinder will have greater volume.
1. While one student holds the cylinder in place measure and fill A with rice.
What is its approximate volume? ___________
2. Slip B over and around A. Lift A up and out of B so the rice begins to fill the
wider cylinder. What occurred once A was completely removed?
________________________
Answer the following:
1. Surface Area of A: __________
2. Surface Area of B: __________
Cylinder Savvy
Student Materials Page 9 of 22
3. Prediction- Which cylinder will have greater volume? ________________
Why? _______________________________________
4. Why do you think the result occurs as it does?
________________________________________________________________
5. Can you sketch 2 cylinders for which the volume would be the same, but the surface
area would be different?
________________________________________________________________
Cylinder Savvy
Student Materials Page 10 of 22
Schematics Needed for Cylinder Savvy Lesson
CD/DVD
Diameter = 12 cm, Thickness (height) = 0.5mm
Mini-CD
Diameter = 80 mm, Thickness (height) = 0.5mm
Cylinder Savvy
Student Materials Page 11 of 22
Disc Storage Websites:
1. www.sleevetown.com/dvd-case.shtml
2. http://www.caselogic.com/search/index.cfm?Ne=100&N=4011+20025939
3. http://www.mediastoragecenter.com/scripts/prodlist.asp?idcategory=237&sortField=price&idsft=757&ntype=DE&
Cylinder Savvy
Student Materials Page 12 of 22
SM-6
Cylinder Savvy: Design 1
Student Instructions
Name ______________________
You are a project designer/engineer hired to make a detailed plan for a CD or DVD
storage container. You will plan three versions of your design and make comparisons
between each one regarding materials needed (surface area) and disc capacity
(volume).
You will need:
• a metric ruler
•
•
a DVD or CD
a mini disc (or metric measurements for one)
Design 1:
1. Sketch a design for holding 12 discs securely. It must be in the shape of a
rectangular prism, triangular prism, cylinder, or other three-dimensional figure
you can calculate the surface area and volume for. You can arrange discs so
that they are in sleeves, stack or pile, snap into place, etc. Be creative. Just
keep in mind how your overall dimensions will be affected by the discs, any
covers or stabilizing materials (sleeves or plastic framing), and open space.
Label length, width, and height of the total structure, as it appears from the
outside.
2. Describe how your disc-holder works. How are discs stored, how does it open
and close, and are there other important characteristics about your design that
makes it unique?
________________________________________________________________
____________________________________________________________
Cylinder Savvy
Student Materials Page 13 of 22
3. What do you predict will be your overall surface area and volume in centimeters?
Surface area prediction: ____________ square centimeters
Disc storage volume prediction: ____________ cubic centimeters
4. Now find the total surface area of Design 1. _______________
5. Find the volume of Design 1. Only calculate the volume for space that would be
used to store discs. _________________
6. Were your predictions close? _________ Why or why not?
___________________________________________________________
7. Do you have any unused space that is a part of your design, but does not contain
discs? If so, find the volume of the extra, unused space.
What percent of your total volume is for actual disc storage? _____________
Why would this be important to consider?
________________________________________________________________
____________________________________________________________
Cylinder Savvy
Student Materials Page 14 of 22
SM-7
Cylinder Savvy: Design 2
Name _______________
The manufacturer wants you to prepare a prototype that is similar in design, but will hold
more discs. Read and complete the following.
1. Sketch a taller version of Design 1, capable of holding 24 discs. Change only
one dimension in your plan, so that you can hold twice as many discs as before.
That is, the outer appearance of your design should only grow in one dimension
– height, length, or width. Label all outer dimensions.
2. How do you predict the change in height will affect your overall surface area and
volume in centimeters? Why?
___________________________________________________________________
___________________________________________________
Surface area prediction: ____________ square centimeters
Disc storage volume prediction: ____________ cubic centimeters
3. Now calculate the total surface area of Design 2. _______________
Cylinder Savvy
Student Materials Page 15 of 22
4. Find
the
volume
of
the
portion
in
Design
2
which
stores
discs.
_________________
What percent of your total volume is for actual disc storage? _______________
Are
there
any
changes
in
your
percentage?
Why
or
why
not?
______________________________________________________________
______________________________________________________________
5. Were your predictions correct about your change in surface area and volume?
____________ If not, why? __________________________
_________________________________________________________
6. What conclusions can you draw about the way surface area changed?
________________________________________________________________
______________________________________________________
7. What
conclusions
can
you
draw
about
the
way
volume
changed?
________________________________________________________________
______________________________________________________
Cylinder Savvy
Student Materials Page 16 of 22
SM-8
Cylinder Savvy: Design 3
Name _______________
Now the manufacturing company has realized there is a market out there for smaller,
mini discs. Find the necessary measurements for a mini disc and then complete the
following.
1. Sketch a narrower version of Design 1, for storing mini discs. Make it the same
height as Design 1. (In other words, take your first design and change only the
measurements needed to affect capacity according to radius/diameter.) Label all
dimensions.
2. Sketch each of the 3 surfaces of your original design below. Then shade the
regions that will still be a part of the new, smaller design. Use this to help you
make your predictions.
3. How do you predict the change in radius/diameter will affect your overall surface
area and volume? Why?
___________________________________________________________
Surface area prediction: ____________ square centimeters
Disc storage volume prediction: ____________ cubic centimeters
4. Find the surface area of Design 3. _______________
5. Find the volume of Design 3. _________________
6. What percent of your total volume is for actual disc storage? ___________
Are there any changes in your percentage? Why or why not?
___________________________________________________________
Cylinder Savvy
Student Materials Page 17 of 22
7. What general conclusions can you make about the way changing height or
diameter affects the surface area of a design?
___________________________________________________________
8. What general conclusions can you make about the way changing height or
diameter
affects
the
volume
used
up
by
discs
in
your
design?
___________________________________________________________
Cylinder Savvy
Student Materials Page 18 of 22
SM-9
Silo: Cylinder Change in Radius
Name _____________________
Extra Practice (following Cylinder Savvy)
With the following problem, you will note how a change in radius can affect both surface
area and volume.
1. Find the surface area and volume of a silo that is 30 feet tall and has a 10 foot
diameter.
Sketch
Surface Area
2 ⋅ πr 2 + 2πr ⋅ h
Volume
πr 2 x h
2. The farmer wants to double the storage space (volume) in his new silo, so he
increases the diameter to 20 feet. Will this work? Sketch the new silo and
calculate the new SA and V.
Sketch
Surface Area
2 ⋅ πr 2 + 2πr ⋅ h
Volume
πr 2 x h
3. How did doubling the radius affect the surface area? Why? Sketch the three
surfaces of the new silo and shade the area that has been added to each
surface.
________________________________________________________________
________________________________________________________________
4. How did it affect the volume? Why?
________________________________________________________________
________________________________________________________________
5. Consider this problem. A circle with a radius of 1 is placed into a larger circle
with radius 2. Compare the areas. How does this problem compare to our silo
situation?
Cylinder Savvy
Student Materials Page 19 of 22
6. Is there a solution to doubling the volume of a silo? What might you be able to
do?
Discuss with a partner two ways you could double the volume of the
original silo. Be sure to include a sketch of each and label all dimensions.
Cylinder Savvy
Student Materials Page 20 of 22
SM-10
Candles: Cylinder Change in Height & Radius
Cylinder Savvy Extension
Name _______________________
With the following problem, you will note how a change in height will affect the surface
area and volume of a cylinder.
1. The Wax Candle Company is planning a new line of candles and figuring out how
best to wrap them. One employee wants to sell candles in 2 sizes, both with a
diameter of 3 inches:
small (3in. X 4 in. tall)
large (3in. X 6 in. tall)
Find the amount of wax needed to create each one (volume) and the
approximate amount of tissue to wrap them (surface area). Record your answers
in the table on number 2. Use 3.14 for pi in your calculations.
2. Wax costs $.10 per cubic inch. Tissue paper costs $.05 per square inch. Find
the cost of making both a small and a large candle in the sizes listed for number
one. Round your answer to the nearest cent.
Volume
(cu. in.)
Surface Area
(sq. in.)
Wax
Cost Tissue
Cost
($.10/cu. in.)
($.05/sq. in.)
Small (3 x 4)
Large (3 x 6)
3. Another employee wants to sell candles that have diameters of 4 inches because
she claims it will save the company money on wax and on tissue for wrapping.
Her designs will be:
small (4in. X 2in. tall)
large (4in. X 4in. tall)
Fill out the table below in order to make comparisons to number 2.
Volume
(cu. in.)
Surface Area
(sq. in.)
Wax
Cost Tissue
Cost
($.10/cu. in.)
($.05/sq. in.)
Small (4 x 2)
Large (4 x 4)
Cylinder Savvy
Student Materials Page 21 of 22
4. Which size, the 3 or 4 inch diameter will save the Wax Candle Company more
money? Why? Be sure to mention both the effect on surface area and volume.
________________________________________________________________
______________________________________________________
5. In your work, you used 3.14 instead of pi to make your calculations. Using pi
would have given a more accurate measurement for surface area and volume.
Why are using 3.14 and rounding your answers to the nearest cent acceptable
for solving this situation? Name a situation for which it would be better to use pi
to find a more accurate answer.
________________________________________________________________
________________________________________________________________
Cylinder Savvy
Student Materials Page 22 of 22