Grade 7 Workbook — Chapter 12: Volume and Surface Area: Prisms and Cylinders

SOL 7.MG.1 · Companion to Textbook Chapter 12

Each page below is one Canva page. Headings are sized for direct paste: page title as H1, section labels as H2. Figures referenced by filename live in ../figures/. Item numbers match the textbook exactly, and every problem uses the same numbers as the textbook problem with the same number.

PAGE 1 — Chapter opener

Chapter 12 · Volume and Surface Area: Prisms and Cylinders

Standard 7.MG.1

In this chapter you will:

• Build and use the volume formula for a right cylinder
• Build and use the surface area formulas for rectangular prisms and right cylinders, starting from nets
• Decide whether a problem is asking for volume or for surface area
• Describe what happens to volume when one dimension is multiplied by $\tfrac14$, $\tfrac13$, $\tfrac12$, 2, 3, or 4
• Describe what happens to surface area when one dimension is multiplied by $\tfrac12$ or 2

Words to know: volume · cubic unit · base area · right cylinder · radius · diameter · surface area · square unit · face · net · rectangular prism · cube · lateral area · circumference · factor

Two rules for every page of this chapter

Pi. Give every answer with $\pi$ twice: exactly in terms of $\pi$, then approximately using $\pi \approx 3.14$.

Units. Volume gets cubic units (cm$^3$, in$^3$, ft$^3$, m$^3$). Surface area gets square units (cm$^2$, in$^2$, ft$^2$, m$^2$). An answer without units is not finished.

The four formulas you will build

$$V_{\text{prism}} = lwh \qquad V_{\text{cylinder}} = \pi r^2 h$$

$$SA_{\text{prism}} = 2lw + 2lh + 2wh \qquad SA_{\text{cylinder}} = 2\pi r^2 + 2\pi r h$$

PAGE 2 — Cubic units and square units

Filling Is Not Covering

FIGURE: fig9-cubic-versus-square-units.png (full width)

Fill in the blanks.

A square unit is ____________ . It measures ____________ , written with a small ______ .

A cubic unit is ____________ . It measures ____________ , written with a small ______ .

Label each unit as a volume unit or an area unit.

Explain. Why does volume need three measurements while area needs only two?

PAGE 3 — Building the cylinder volume formula

12.1 Volume of a Right Cylinder

FIGURE: fig2-cylinder-layers.png (full width)

Complete the derivation.

For any prism or cylinder, volume $= (\text{area of the } \underline{\hspace{2cm}}) \times (\underline{\hspace{2cm}})$, or $V = B \times h$.

The base of a cylinder is a ____________ , and the area of a circle is $A = \underline{\hspace{2cm}}$.

Substituting: $V = \underline{\hspace{3cm}}$

$$\boxed{V = \pi r^2 h}$$

FIGURE: fig1-cylinder-labeled.png (right half of page)

Label the figure. Write $r = \underline{\hspace{1.5cm}}$ and $h = \underline{\hspace{1.5cm}}$ on the arrows, then find the volume.

$V = \pi(\underline{\hspace{1cm}})^2(\underline{\hspace{1cm}}) = \underline{\hspace{1.5cm}}\pi$ cm$^3 \approx \underline{\hspace{2cm}}$ cm$^3$

Two traps. Circle the correct choice.

In $\pi r^2$, you ( square the radius first / multiply by $\pi$ first ).

If a problem gives the diameter, you must ( double it / halve it ) to get the radius.

PAGE 4 — Guided practice 12.1

Volume of a Cylinder · Guided Practice

Formula frame: $V = \pi r^2 h$ · units are cubic

1. Radius $2$ cm, height $7$ cm.

$V = \pi(\underline{\hspace{1cm}})^2(\underline{\hspace{1cm}}) = \underline{\hspace{1.5cm}}\pi$ cm$^3 \approx \underline{\hspace{2cm}}$ cm$^3$

2. Radius $6$ in, height $4$ in.

$V = \underline{\hspace{6cm}}$

3. Diameter $10$ m, height $3$ m. First: $r = \underline{\hspace{1cm}}$ m.

$V = \underline{\hspace{6cm}}$

4. Radius $1$ ft, height $12$ ft.

$V = \underline{\hspace{6cm}}$

5. Base area $20\pi$ cm$^2$, height $6$ cm. Use $V = Bh$.

$V = \underline{\hspace{6cm}}$

PAGE 5 — Independent practice 12.1

Volume of a Cylinder · Independent Practice

Complete the table. Give each volume exactly in terms of $\pi$ and then with $\pi \approx 3.14$.

10. A cylinder has radius $5$ in and volume $100\pi$ in$^3$. Find its height.

$100\pi = \pi(\underline{\hspace{1cm}})^2 h \ \Rightarrow \ 100\pi = \underline{\hspace{1cm}}\pi h \ \Rightarrow \ h = \underline{\hspace{1.5cm}}$ in

11. Application. A soup can is a right cylinder with radius $4$ cm and height $11$ cm. How much soup does it hold when full?

12. Reasoning. Explain why the volume of a cylinder is measured in cubic units. Use $\pi r^2$ and $h$ in your explanation.

PAGE 6 — Exit ticket 12.1

Exit Ticket · Lesson 12.1

Name: ________________________ Date: ____________

13. Radius $2$ in, height $10$ in. $V = \underline{\hspace{5cm}}$

14. Diameter $6$ cm, height $5$ cm. $V = \underline{\hspace{5cm}}$

15. Base area $30\pi$ m$^2$, height $4$ m. $V = \underline{\hspace{5cm}}$

16. A cylinder has diameter $12$ cm and height $2$ cm. Dana wrote $V = \pi(12)^2(2) = 288\pi$ cm$^3$. Find her error and give the correct volume, exactly and approximately.

Her error: _______________________________________________

Correct volume: _______________________________________________

PAGE 7 — Unfolding a box

12.2 Surface Area of a Rectangular Prism

FIGURE: fig3-prism-net.png (full width)

Net-folding work. The prism above is $l = 5$ cm, $w = 3$ cm, $h = 4$ cm. Label each face of the net with its name and its area.

Write the formula from your table.

$$SA = 2\underline{\hspace{1cm}} + 2\underline{\hspace{1cm}} + 2\underline{\hspace{1cm}}$$

A cube is a special case. All six faces are $s \times s$, so $SA = \underline{\hspace{1.5cm}}$ .

Fold check. How many rectangles are in the net of a rectangular prism? ______ How many matching pairs? ______

PAGE 8 — Guided practice 12.2

Surface Area of a Prism · Guided Practice

Formula frame: $SA = 2(lw + lh + wh)$ · units are square

17. $4$ cm by $3$ cm by $2$ cm.

$SA = 2(\underline{\hspace{1cm}} + \underline{\hspace{1cm}} + \underline{\hspace{1cm}}) = \underline{\hspace{1.5cm}}$ cm$^2$

18. Cube with edge length $5$ in.

$SA = 6(\underline{\hspace{1cm}})^2 = \underline{\hspace{1.5cm}}$ in$^2$

19. $7$ ft by $5$ ft by $2$ ft. $SA = \underline{\hspace{5cm}}$

20. $10$ m by $6$ m by $3$ m. $SA = \underline{\hspace{5cm}}$

21. A net is made of two $8 \times 3$ rectangles, two $8 \times 2$ rectangles, and two $3 \times 2$ rectangles, in inches.

$SA = 2(\underline{\hspace{1cm}}) + 2(\underline{\hspace{1cm}}) + 2(\underline{\hspace{1cm}}) = \underline{\hspace{1.5cm}}$ in$^2$

PAGE 9 — Independent practice 12.2

Surface Area of a Prism · Independent Practice

Computation table.

26. A prism is $7$ cm by $4$ cm by $3$ cm. List the three different face shapes in its net, how many of each, and the surface area.

27. Application. An aquarium with no lid is $8$ ft long, $4$ ft wide, and $3$ ft tall. Count the bottom and the four sides only.

bottom $= \underline{\hspace{1.5cm}}$ · front and back $= \underline{\hspace{1.5cm}}$ · two sides $= \underline{\hspace{1.5cm}}$ · total $= \underline{\hspace{1.5cm}}$ ft$^2$

28. Reasoning. Explain why surface area is measured in square units and not cubic units, and how a net makes that obvious.

PAGE 10 — Exit ticket 12.2

Exit Ticket · Lesson 12.2

Name: ________________________ Date: ____________

29. $5$ in by $4$ in by $3$ in. $SA = \underline{\hspace{4cm}}$

30. Cube with edge length $4$ cm. $SA = \underline{\hspace{4cm}}$

31. A net of two $6 \times 2$, two $6 \times 5$, and two $2 \times 5$ rectangles, in feet. $SA = \underline{\hspace{4cm}}$

32. For a prism that is $5$ in by $4$ in by $3$ in, Owen wrote $SA = 2(20) + 15 + 12 = 67$ in$^2$.

His error: _______________________________________________

Correct surface area: _______________________________________________

PAGE 11 — Unrolling a can

12.3 Surface Area of a Right Cylinder

FIGURE: fig4-cylinder-net.png (full width)

Net-folding work. Cut apart a cylinder and you get three pieces. Name them and give each area.

Why is the rectangle that wide? The label travels once around the circle, and the distance once around a circle is its ____________ , which equals ______________ .

Add the pieces.

two circles $= \underline{\hspace{2cm}}$ $+$ rectangle $= \underline{\hspace{2cm}}$

$$\boxed{SA = 2\pi r^2 + 2\pi r h}$$

Name the parts. $2\pi r^2$ is the area of the ____________ . $2\pi r h$ is the ____________ area, which is what a label covers.

PAGE 12 — Guided practice 12.3

Surface Area of a Cylinder · Guided Practice

Formula frame: $SA = 2\pi r^2 + 2\pi r h$ · units are square

33. Radius $1$ m, height $4$ m.

$SA = 2\pi(\underline{\hspace{1cm}})^2 + 2\pi(\underline{\hspace{1cm}})(\underline{\hspace{1cm}}) = \underline{\hspace{1cm}}\pi + \underline{\hspace{1cm}}\pi = \underline{\hspace{1.5cm}}\pi$ m$^2 \approx \underline{\hspace{2cm}}$ m$^2$

34. Radius $5$ cm, height $2$ cm. $SA = \underline{\hspace{5cm}}$

35. Diameter $8$ in, height $3$ in. First $r = \underline{\hspace{1cm}}$ in. $SA = \underline{\hspace{5cm}}$

36. Lateral area only: radius $6$ ft, height $5$ ft. $2\pi r h = \underline{\hspace{4cm}}$

37. Radius $2$ cm, height $10$ cm. $SA = \underline{\hspace{5cm}}$

PAGE 13 — Independent practice 12.3

Surface Area of a Cylinder · Independent Practice

Computation table.

41. Lateral area only: radius $2$ ft, height $7$ ft. $\underline{\hspace{5cm}}$

42. A bucket with no lid has radius $5$ in and height $6$ in. Count one circle and the side.

one circle $= \underline{\hspace{1.5cm}}$ $+$ side $= \underline{\hspace{1.5cm}}$ $=$ $\underline{\hspace{2cm}}$ in$^2 \approx \underline{\hspace{2cm}}$ in$^2$

43. Application. A label wraps once around a can of radius $4$ cm and height $10$ cm, covering the side only.

Area of label $= \underline{\hspace{4cm}}$

The flat rectangle it is cut from is ____________ cm wide by ____________ cm tall.

44. Reasoning. Why is the width of the rectangle $2\pi r$? What would go wrong if you used $r$ instead?

PAGE 14 — Exit ticket 12.3

Exit Ticket · Lesson 12.3

Name: ________________________ Date: ____________

45. Radius $2$ cm, height $3$ cm. $SA = \underline{\hspace{5cm}}$

46. Diameter $12$ in, height $5$ in. $SA = \underline{\hspace{5cm}}$

47. Lateral area only: radius $1$ m, height $9$ m. $\underline{\hspace{5cm}}$

48. A cylinder has radius $3$ cm and height $8$ cm. Malik answered $48\pi$ cm$^2$.

Part of the net he forgot: _______________________________________________

Correct surface area: _______________________________________________

PAGE 15 — Inside or outside?

12.4 Volume or Surface Area?

FIGURE: fig5-volume-or-surface-area.png (full width)

One question decides it. Is the problem about the ____________ or the ____________ ?

Complete the sorting chart.

Units are the double check. If a question asks how much paint and your answer says ft$^3$, you found the ____________ instead of the ____________ .

The four formulas. Write each one from memory.

$V_{\text{prism}} = \underline{\hspace{3cm}}$ $V_{\text{cylinder}} = \underline{\hspace{3cm}}$

$SA_{\text{prism}} = \underline{\hspace{4cm}}$ $SA_{\text{cylinder}} = \underline{\hspace{4cm}}$

PAGE 16 — Guided practice 12.4

Volume or Surface Area · Guided Practice

For each item, first circle V or SA, then compute.

49. Wrapping paper to cover a closed box $8$ in by $5$ in by $2$ in. V / SA

Answer: _______________________________________________

50. Water held by a cylindrical tank, radius $3$ m and height $10$ m. V / SA

Answer: _______________________________________________

51. Cereal that fills a box $10$ in by $6$ in by $3$ in. V / SA

Answer: _______________________________________________

52. Foil to cover a closed cylinder completely, radius $1$ in and height $6$ in. V / SA

Answer: _______________________________________________

53. Concrete that fills a cylindrical post hole, radius $1$ ft and depth $4$ ft. V / SA

Answer: _______________________________________________

PAGE 17 — Independent practice 12.4

Volume or Surface Area · Independent Practice

54. Write volume or surface area and name the units.

Same solid, two questions. A box is $12$ in by $3$ in by $2$ in.

55. Gift wrap to cover it exactly: _______________________________________________

56. Rice that fills it: _______________________________________________

Same solid, two questions. A cylinder has radius $2$ in and height $9$ in.

57. Juice it holds: _______________________________________________

58. Metal to make it, top and bottom included: _______________________________________________

59. Application. An open-top cylindrical trough has radius $2$ ft and height $5$ ft.

a) Water it holds: _______________________________________________

b) Metal for the bottom and curved side: _______________________________________________

60. Reasoning. A classmate hands you the answer $150$ cm$^2$ but forgot the question. What kind of question was it, and how do you know?

PAGE 18 — Exit ticket 12.4

Exit Ticket · Lesson 12.4

Name: ________________________ Date: ____________

61. Volume or surface area: how much wrapping paper covers a shoebox? ______________

62. A cereal box is $8$ in by $2$ in by $12$ in. How much cereal fills it? $\underline{\hspace{4cm}}$

63. A closed cardboard tube is a cylinder with radius $3$ in and height $4$ in. How much cardboard? $\underline{\hspace{4cm}}$

64. A student says a room needs $96$ cm$^3$ of paint. Explain what is wrong without doing arithmetic.

PAGE 19 — One dimension, one factor

12.5 Changing One Dimension: Volume

FIGURE: fig6-double-height-volume.png (full width)

Compute before and after.

$V_{\text{after}} \div V_{\text{before}} = \underline{\hspace{1cm}}$ , so the volume was multiplied by ______ .

FIGURE: fig8-half-length-volume.png (full width)

$V_{\text{after}} \div V_{\text{before}} = \underline{\hspace{1cm}}$ , so the volume was multiplied by ______ .

State the pattern.

Multiplying one measured attribute of a rectangular prism by a factor multiplies the volume by ______________ .

Why? $V = l \times w \times h$. If the height becomes $2h$, then $V_{\text{new}} = l \times w \times 2h = \underline{\hspace{1cm}} \times (lwh)$.

Caution. This rule is for changing ______ attribute. Allowed factors for volume: $\tfrac14$, $\tfrac13$, $\tfrac12$, 2, 3, 4.

PAGE 20 — Guided practice 12.5

Changing One Dimension: Volume · Guided Practice

Items 65–68 all start from the same prism: $8$ cm by $3$ cm by $4$ cm, volume $= \underline{\hspace{1.5cm}}$ cm$^3$.

Use the shortcut. New volume $=$ old volume $\times$ factor.

69. $V = 200$ in$^3$; width $\times 4$. New $V = \underline{\hspace{1cm}} \times \underline{\hspace{1cm}} = \underline{\hspace{2cm}}$ in$^3$

70. $V = 72$ m$^3$; length $\times \tfrac13$. New $V = \underline{\hspace{3cm}}$ m$^3$

PAGE 21 — Independent practice 12.5

Changing One Dimension: Volume · Independent Practice

74. A prism has volume $90$ m$^3$. One attribute is multiplied by $3$. New $V = \underline{\hspace{3cm}}$

75. A prism has volume $48$ in$^3$. One attribute is multiplied by $\tfrac12$. New $V = \underline{\hspace{3cm}}$

76. Application. A shipping box is $9$ in by $4$ in by $5$ in. The height is doubled.

$V$ before $= \underline{\hspace{2cm}}$ $V$ after $= \underline{\hspace{2cm}}$ How many more cubic inches? $\underline{\hspace{2cm}}$

77. Reasoning. Use $V = lwh$ to explain why multiplying just the width by $3$ multiplies the volume by $3$. Then explain why multiplying all three dimensions by $3$ is a different situation.

PAGE 22 — Exit ticket 12.5

Exit Ticket · Lesson 12.5

Name: ________________________ Date: ____________

78. $6$ cm by $5$ cm by $2$ cm; height $\times 3$. New $V = \underline{\hspace{3cm}}$

79. $6$ cm by $5$ cm by $2$ cm; length $\times \tfrac12$. New $V = \underline{\hspace{3cm}}$

80. $V = 100$ ft$^3$; one attribute $\times 4$. New $V = \underline{\hspace{3cm}}$

81. Rosa says doubling the length makes the volume $8$ times as large.

Her error: _______________________________________________

Correct factor: ______

PAGE 23 — A rule that does not carry over

12.6 Changing One Dimension: Surface Area

FIGURE: fig7-double-height-surface-area.png (full width)

Compute before and after for the prism $4$ cm by $3$ cm by $2$ cm with the height doubled.

Twice the original surface area would be $2 \times \underline{\hspace{1cm}} = \underline{\hspace{1cm}}$ cm$^2$. The actual new surface area is $\underline{\hspace{1cm}}$ cm$^2$, so the surface area ( did / did not ) double.

Sort the six faces.

Say why. Part of the surface ______________ and part of it ______________ , so the total grows by ( more / less ) than a factor of 2.

Allowed factors for surface area: $\tfrac12$ and $2$ only.

PAGE 24 — Guided practice 12.6

Changing One Dimension: Surface Area · Guided Practice

Items 82–85 all start from the same prism: $5$ cm by $4$ cm by $3$ cm, $SA = 94$ cm$^2$.

85. In item 82, which two faces did not change, and what is the area of each?

86. A cube has edge length $4$ in and its height is multiplied by $\tfrac12$.

$SA$ before $= \underline{\hspace{2cm}}$ $SA$ after $= \underline{\hspace{2cm}}$

PAGE 25 — Independent practice 12.6

Changing One Dimension: Surface Area · Independent Practice

91. A cube has edge length $4$ in and its height is multiplied by $2$.

$SA$ before $= \underline{\hspace{2cm}}$ $SA$ after $= \underline{\hspace{2cm}}$ Did it double? ______ Actual change: $\underline{\hspace{2cm}}$

92. Application. A shipping box is $12$ in by $6$ in by $4$ in. The maker multiplies the height by $\tfrac12$.

Cardboard before $= \underline{\hspace{2cm}}$ after $= \underline{\hspace{2cm}}$ saved per box $= \underline{\hspace{2cm}}$

93. Reasoning. Explain why doubling one attribute doubles the volume but not the surface area. Name the faces that change and the faces that do not.

PAGE 26 — Exit ticket 12.6

Exit Ticket · Lesson 12.6

Name: ________________________ Date: ____________

Items 94–96 use the prism $6$ cm by $4$ cm by $2$ cm.

94. Height $\times 2$. $SA$ before $= \underline{\hspace{2cm}}$ $SA$ after $= \underline{\hspace{2cm}}$

95. Starting again from $6 \times 4 \times 2$, length $\times \tfrac12$. $SA$ before $= \underline{\hspace{2cm}}$ $SA$ after $= \underline{\hspace{2cm}}$

96. $V$ of $6 \times 4 \times 2 = \underline{\hspace{1.5cm}}$ ; after height $\times 2$, $V = \underline{\hspace{1.5cm}}$ . Volume factor $= \underline{\hspace{1cm}}$ . Did the surface area change by that same factor? ______

97. In one or two sentences, explain why volume and surface area respond differently when one attribute is doubled.

PAGE 27 — Review Parts A and B

Chapter 12 Review · Volume and Surface Area Formulas

Vocabulary check. volume · cubic unit · base area · right cylinder · radius · diameter · surface area · square unit · face · net · rectangular prism · cube · lateral area · circumference · factor

Part A — Volume of right cylinders (7.MG.1a). Exact, then $\pi \approx 3.14$.

98. $r = 4$ cm, $h = 5$ cm $\underline{\hspace{5cm}}$

99. $d = 10$ in, $h = 6$ in $\underline{\hspace{5cm}}$

100. $r = 2$ m, $h = 11$ m $\underline{\hspace{5cm}}$

101. $r = 3$ ft, $V = 63\pi$ ft$^3$. Find $h$. $h = \underline{\hspace{2cm}}$

102. Application. A cylindrical drum has $r = 3$ ft and $h = 4$ ft. Water held: $\underline{\hspace{4cm}}$

Part B — Surface area (7.MG.1b).

103. Prism $7$ cm by $3$ cm by $2$ cm $\underline{\hspace{4cm}}$

104. Cube, edge $8$ in $\underline{\hspace{4cm}}$

105. Cylinder $r = 5$ m, $h = 3$ m $\underline{\hspace{5cm}}$

106. Cylinder $d = 4$ in, $h = 10$ in $\underline{\hspace{5cm}}$

107. Net of two $9 \times 5$, two $9 \times 2$, two $5 \times 2$ rectangles (feet) $\underline{\hspace{4cm}}$

108. Lateral area only, cylinder $r = 6$ cm, $h = 2$ cm $\underline{\hspace{5cm}}$

PAGE 28 — Review Parts C, D, and E

Chapter 12 Review · Deciding and Scaling

Part C — Volume or surface area? (7.MG.1c)

109. Write volume or surface area and the units.

A box is $10$ in by $4$ in by $3$ in.

110. Paint to cover its outside: $\underline{\hspace{4cm}}$

111. Sand that fills it: $\underline{\hspace{4cm}}$

112. Application. An open-top cylindrical barrel has $r = 2$ ft and $h = 6$ ft.

a) Water held: $\underline{\hspace{4cm}}$ b) Metal for bottom and curved side: $\underline{\hspace{4cm}}$

Part D — Changing one attribute: volume (7.MG.1d)

115. $V = 60$ m$^3$; one attribute $\times \tfrac12$. New $V = \underline{\hspace{3cm}}$

116. Application. A grain bin is $6$ ft by $5$ ft by $4$ ft; the width is doubled.

before $= \underline{\hspace{2cm}}$ after $= \underline{\hspace{2cm}}$ more $= \underline{\hspace{2cm}}$

Part E — Changing one attribute: surface area (7.MG.1e)

PAGE 29 — Review Part F

Chapter 12 Review · Mixed Application and Reasoning

120. A closed cylindrical can has $r = 3$ in and $h = 10$ in.

a) Volume: $\underline{\hspace{5cm}}$

b) Surface area: $\underline{\hspace{5cm}}$

c) Which answer tells how much juice it holds? ______________

121. Explain why volume is reported in cubic units and surface area in square units. Use $V = lwh$ and $SA = 2lw + 2lh + 2wh$.

122. A student claims doubling the height of a prism doubles both the volume and the surface area. Use $4$ cm by $3$ cm by $2$ cm to test both claims.

Which claim is right? ______________ Which is wrong? ______________

123. Explain how the net of a cylinder shows where each term of $SA = 2\pi r^2 + 2\pi r h$ comes from.

124. Application. A cylindrical can has $r = 4$ cm and $h = 10$ cm.

a) How much it holds: $\underline{\hspace{5cm}}$

b) Label covering the side only: $\underline{\hspace{5cm}}$

c) Which answer is in cm$^3$, and why? _______________________________________________