Unit 6 – Area, and Volume
Lesson 2 – Surface Area
Black – Solids and Surface Area
1. What is the number of square inches in the total surface
area of the resulting figure when a 2” x 2” x 2” cube is
removed from a 4” x 4” x 4” cube as shown?
2. A rectangular table is 45 inches wide and
64 inches long. A tablecloth hangs over
the table 10 inches at each edge. What is
the number of square inches in the area
of the tablecloth?
3. Surface Area. Some unit cubes are placed side by side to form rectangular prisms.
Determine the surface area for prism formed from one, two, three, four, or five
cubes. Write an expression for finding the surface area of a prism with n cubes.
4. Mailbox Sliders Some snowboarding parks
have a feature shaped like an elongated
mailbox and called a mailbox slider.
Snowboarders use their boards to jump
onto the mailbox slider and slide along its
top surface. A mailbox slider is composed
of a half cylinder and rectangular prism.
Find the surface area of the mailbox slider
shown. Round your answer to the nearest
square inch.
8 in.
4 in.
180 in.
8 in.
Not draw
To scale
5. A timber company sells ‘half-rounds’. They are used to build retaining walls.
1
Unit 6 – Area, and Volume
Lesson 2 – Surface Area
6. A dog is tied to the middle of one side of barn. The dog ruins the grass wherever it
can reach. The dog is on a 26 foot leach and the side of the barn where he is tied is
26 feet long. (The farthest out the dog can reach is 26 feet.) What is the area of
the grass that the dog will ruin.
20 feet
Barn
26 feet
7. The area of the outside square is 81 square feet.
What is the area of the shaded Isosceles triangle
that is inside the small square?
8. Find the area of the shaded region. The
diameter of the circle is 10 feet.
9. Wendy is growing corn in a 100’ by 100’ farm field. Some friends decided to play a
trick on Wendy to make it appoear as though aliens visited her farm. They fastened
one end of a 20 foot board to the ground and rotated it in complete circle. In the
process they destroyed all the corn plants within that circle. Wendy was going to
make $1500 profit from her corn, but because some of th field was damaged, she will
make less money. How much money should Wendy expect to make now?
10. What is the area of the shaded part inside the 10 foot square?
2
Unit 6 – Area, and Volume
Lesson 2 – Surface Area
11. The circle is perfectly centered on top of a rectangle that is 20 inches long. Lines AC
and BD are both diameters of the circle and are equal to
shaded part?
(Worth three Einsteins)
5 . What is the areae
12. How Does Your Garden Grow?
The students at Highland Middle School plan to plant a small garden near their school
yard pond. The students agree that the garden will be planted in a square, and it will
measure three meters on each side. (See diagram below.) They also decide that they
will divide the garden into nine one square meter plots. The students select three
vegetables to plant, and they agree that they will plant the following amounts of each
vegetable:
•
•
•
Eggplant: 4 square meters
Broccoli: 3 square meters
Red Peppers: 2 square meters
Before the students each submit a plan for the garden, they decide that they should
make the garden as easy to care for as possible. To do this everyone agrees that
they should plant all of the squares of each type of vegetable side to side. The
students want to avoid any arrangement that has plots of one type of vegetable not
meeting at all or only meeting at their corners. For example:
Acceptable
Arrangement
Unacceptable
Arrangement
Each student is then asked to submit one plan for planting the vegetables. The
students and the teacher agree that the most frequently submitted design will be
the plan that those planting the garden will use. One plan looks like the one below.
What is the fewest number of plans that students could submit to ensure that at
least one plan gets at least two votes? In order to receive credit for answering this
problem, make sure you explain the process you used to arrive at your solution.
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3
Unit 6 – Area, and Volume
Lesson 2 – Surface Area
Solutions
1. The three full faces with no piece removed have area 4² = 16 in² each. The three
small faces left after the cube is removed each have area 2² = 4 in². The area of
each of the L-shaped faces are composed of a 4 x 2 = 8 in²rectangle and a 2 x 2 = 4
in², so each of them has area 8 + 4 = 12 in². The total surface area, then, is 3(16) +
3(4) + 3(12) = 96 in². Note that this is the same surface area as the original cube,
because the area of each removed face is equal to the area of each new face.
2. The cloth hangs over 10 inches on all four edges, so the width of the cloth is 10 + 45
+ 10 = 65 inches. Likewise, the length is 10 + 64 + 10 = 84 inches. The area is 84 x 65
= 5460 square inches.
3. Surface Area. 4n + 2. For the first five prism, the area are 6, 10, 14, 18, and 22.
Each prism has two square faces on opposite ends. Each cube contributes four square
units to the surface area contained in the remaining faces; hence 4n + 2.
4. 6760 in.²
5. 0.712 m²
6. 1326.65 square feet
The area of the grass that is ruined is half of a circle with a 26 foot radius. In
addition, as the leash catches on the corner of the barn, two quarter circles are
formed that each have a radius of 13 feet.
Area of large half circle:
(3.14 x 26 x 26) ÷ 2 = 1061.32 square feet.
Area of quarter circles:
(3.14 x 13 x 13) ÷ 4 = 132.665 square feet.
Total area: 1061.32 + 132.665 = 1326.65 square feet
7. 20.25 square feet
The length of each side of the large square is 9 inches.
8. 25 square feet
The area of the rectangle: 5 x 10 = 50 square feet
The area of the upper half circle: 3.14 x 5 x 5 x .5 = 29.25 square feet
Total area: 50 + 39.25 = 89.25 square feet
The area of the white triangle: 10 x .5 x .5 = 25 square feet
Area of shaded part: 89.25 – 39.25 – 25 = 25 square feet
9. $1311.60
Wendy’s profit per square foot: $1500 ÷ 10,000 = $.15 per square foot.
Part destroyed: 3.14 x 20 x 20 = 1256 square feet
Lost money: 1256 x $.15 = $188.40
Wendy’s projected profits: $1500 - $188.40 = $1311.60
4
Unit 6 – Area, and Volume
Lesson 2 – Surface Area
10. 57 square feet
The area of one white section can be found by
finding the area of the quarter circle and
subtracting it from the total area of the
square.
Area of square: 100 square feet
Area of circle: 3.14 x 10 x 10 = 314 square feet
Area of ¼ circle: 314 ÷ 4 = 78.5 square feet
Area of line white part: 100 – 78.5 = 21.5 square feet
Area of the shaded part: 100 – 43 = 57 square feet
2 white parts: 43 square feet
11. 67.876 square inches
B
C
50
The radius of the circle is
2
50
50
The area of the circle is
x
x 3.14 =
2
2
A
D
50
x 3.14 = 12.5 x 3.14 = 39.25 square inches.
4
The area of a quarter of the circle is 39.25 ÷ 4 = 9.8125 square inches
50
x
2
The area of each of the 4 triangles is
50
x .5 = 12.5 x 5 = 6.25 square
2
inches.
AB (width of the rectangle) can be found using the Pythagorean Theorem.
50
50
50
50
x
+
x
= c²
c²= 25 c = 5
2
2
2
2
Rectangle’s area = 5 x 20 = 100 sq. in.
Area of rectangle (100) – 2 triangles (12.5) – 2 quarter circles (19.625) = 67.875
square inches.
12. 49 different students would need to turn in plot plans.
The first thing I did was see how many different plot configurations I could come up
with. I discovered 6, not counting rotational or mirror images. I started placing the
Eggplants first, because it has the largest area needing to be fitted into the plot
plan. I found 3 different patterns, and one of those I could place differently making
4 configurations. I then looked at the Broccoli. and with the Eggplant in an 'L'
configuration at an edge, there were 3 different placements of the 3 squares.
E =Eggplant
EEP
EEB
EBP
EBB
EBB
EPP
B= Broccoli P= Red
EEB
BEE
EBB
EPB
BEP
EEB
EPB
BEP
EPP
Peppers
EEB
EEB
PPB
Next I have to consider the mirror image and the rotation of both standard
placement and mirror image patterns. That makes 8 variations on each of the 6 above
patterns for a total of 6 x8 =48 different plot plans. The fewest plot plans needed
5
Unit 6 – Area, and Volume
Lesson 2 – Surface Area
to have at least two of one type would be one more than the number of different
patterns or 48 +1 =49.
Ps. It seems like the square pattern would make weeding the center block difficult
without walking into the plantings.
6
Unit 6 – Area, and Volume
Lesson 2 – Surface Area
Bibliography Information
Teachers attempted to cite the sources for the problems included in this problem set. In some cases,
sources may not have been known.
Problems
Bibliography Information
12
The Math Forum @ Drexel
(http://mathforum.org/)
6 - 11
Zaccaro, Edward. Challenge Math (Second
Edition): Hickory Grove Press, 2005.
5
Barton, David. Beta Mathematics Homework
Book. ISBN 978-1-4425-0017-4. Pearson
Education New Zealand, 2000.
7