M6M2. Students will use appropriate units of measure for finding length, perimeter,
area and volume and will express each quantity using the appropriate unit.
b. Select and use units of appropriate size and type to measure length, perimeter,
area, and volume.
M6M4. Students will determine the surface area of solid figures (right rectangular
prisms and cylinders).
a. Find the surface area of right rectangular prisms and cylinders using
manipulatives and constructing nets.
b. Compute the surface area of right rectangular prisms and cylinders using
formulae.
d. Solve application problems involving surface area of right rectangular prisms
and cylinders.
M6P2. Students will reason and evaluate mathematical arguments.
a. Recognize reasoning and proof as fundamental aspects of mathematics
M6P3. Students will communicate mathematically.
a. Organize and consolidate their mathematical thinking through communication.
b. Communicate their mathematical thinking coherently and clearly to peers,
teachers, and others.
c. Analyze and evaluate the mathematical thinking and strategies of others.
Surface Area of solid figures
Questions:

How do you find the surface areas of rectangular prisms and cylinders?
Launch:
Find the area for the shapes below:
10m
3.5m
6cm
Area: ___________
Area:___________
1. What is the formula that you used to find the area of the rectangle?
2. What is the formula that you used for the area of the circle?
3. What type of unit is area measured in? Why?
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Investigation 1
Rectangular Prisms
Look at the net to answer the following questions. (You may work with group members.)
5 cm
c
b
3cm
a
c
3cm
4cm
b
a
4cm
5cm
What is the area of a rectangle labeled “a”? (Since there are two of them, what
would be an expression to represent that area?)
1.
Area of a = _________
Expression = ________
2. What is the area of a rectangle labeled “b”? (Since there are two of them, what
would be an expression to represent that area?)
Area of b = _________
Expression = ________
3.What is the area of a rectangle labeled “c”? (Since there are two of them, what
would be an expression to represent that area?)
Area of c = _________
Expression = ________
4. If you added up the area of all the rectangles (6), what would your total surface
area be?
5. Look at the expressions you wrote for #1-3. How could you write a formula using
these expressions to explain how you found the total area of the figure?
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6. Revise your formula and use length, width, and height to describe the surface area
of the figure. (Instead of your a,b,c expressions, use l,w,h)
Formula = ___________________
7. If the rectangular prism below represents the net above, label the length, width, and
height.
8. Substitute these values into your new formula (using l,w,h) and see if this matches
your answer for #4. Did it work? Why?
9.Compare your formula with your group members. Did you derive the same
formula? If not, exchange ideas until coming to a consensus.
Conclusion:
Based on the investigation, do you think you could you use the same formula on all
rectangular prisms to find the surface area?
Would your formula deviate if you were finding the surface area of a cube? Why?
How could you write this altered formula?
What would happen to the surface area of the rectangular prism if you doubled the length,
width, and height?
With your group members, create two problems to try out your hypothesis of doubling
these measures. Were you correct? Explain.
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Practice
Using your derived formula, calculate the surface areas of the following:
2.6m
1.
14m
4m
2.
8 ½ mm
10mm
12mm
3. Maria wants to wrap a box with a length and width of 5 inches and a height of 8
inches. She only has 200 in2 of wrapping paper. Does she have enough? How do
you know?
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Investigation 2
Cylinders
Look at the net to answer the following questions. (You may work with group members)
8m
6m
13m
1. What is the area of the rectangle?
2. What are the areas of the circles? (Remember to use πr for each circle area. There
are also two of them.)
2
3. If you added up the area of the rectangle and circles, what would your total surface
area be?
4. How could you write a formula for the total surface area of the cylinder?
Hint for formula revision: The length of the rectangle is actually the circumference of the
circle. (pi times diameter or pi times 2 radii.)
h
2Πr
Look at the cylinder below and think about the formula you just wrote. Does it make
sense? Are all surfaces areas accounted for? (top, bottom, and rounded surface)
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Conclusion:
Explain, in generalized terms, the process used for deriving the formulas for the surface
area of solid figures.
Practice:
Determine the surface areas of the following cylinders using your derived formula.
1.
r =2.2ft
11ft
d=17mm
2.
4mm
3. Johnny wants to paint a round trashcan with a diameter of 16 inches and height of
30 inches. What is the total surface area that Johnny has to paint? (Hint: Johnny’s
trashcan does not have a top.)
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Closure:
What are the similarities and differences between finding the surface area of a cylinder
and a rectangular prism?
Why does one formula not work on the other?
Homework:
1. A quarter is really a very short cylinder. It's height is about 1 mm, and it's
diameter is about 24 mm. Find the surface area of a quarter.
let pi = 3.14
2. If a can having a diameter of 4 units is doubled how does the surface area of the
new can compare to the surface area of the original can? The height of the
original can is 8 units and is not changed in creating the new can.
(let pi = 3.14)
3. A cylindrical tube needs to be painted on the inside as well as on the outside. It is
open on both ends. If the diameter of the tube is 18 cm, and it is 110 cm in
length, what is the total surface area of this tube which will be painted?
(let pi = 3.14)
4. Martin wants to paint two columns in his basement. (They are long rectangular
prisms.) The length and width are both 6 inches. The heights are 10 feet. What is
the total surface area that Martin will have to paint? (Hint: Think about columns
that are attached. You may not have to use all the measurements you were given.)
In problem number four, what parts of the formula did you have to leave out? Why?
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References
DeRitter, P. (2001). Oswego City School District. New York State Intermediate Test
Prep Center. Retrieved from: http://www.studyzone.org/mtestprep/math8/g/
cylsurfareaprac.cfm on June 30, 2008.
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