Geometry Workbook — Chapter 17: Changing Dimensions

SOL G.DF.2 (a, b, c) · Companion to Textbook Chapter 17

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/. Every numbered item is the same problem as in the textbook, so one answer key serves both. Item numbers run continuously from 1 to 98.

PAGE 1 — Chapter opener

Chapter 17 · Changing Dimensions

Standard G.DF.2 (a, b, c)

In this chapter you will:

• Describe how uniformly scaling every dimension of a figure by $k$ affects its perimeter or circumference, and its area
• Describe how uniformly scaling every dimension of a solid by $k$ affects its surface area and volume
• Describe how changing only ONE dimension affects volume and surface area differently
• Recover a scale factor from a given change in area, surface area, or volume
• Solve problems in context involving a change in one or more dimensions

Words to know: scale factor · uniform scaling · perimeter · circumference · surface area · volume · derived measure · square root · cube root

Conventions: scale factor is always $k$. Length-like measures scale by $k$; area-like measures (area, surface area) scale by $k^2$; volume scales by $k^3$ — but only when EVERY dimension changes by the same $k$. Changing only one dimension breaks that pattern for surface area, which must be recomputed directly.

PAGE 2 — The whole chapter on one page

One Pattern, Three Exponents

This table holds only when EVERY dimension changes by the same $k$. Change just one dimension, and only the measures whose formula contains it move — surface area almost never moves by a clean power in that case, and must be recomputed.

Before every problem:

• Is every dimension changing by the same $k$, or only one?
• Am I going forward (given $k$ or dimensions, find the new measure) or backward (given a ratio of a measure, find $k$)?

PAGE 3 — A rectangle scaled

17.1 Two Rules From One Idea

FIGURE: fig1-rectangle-scaled.png (full width)

Fill in.

$$P = 2(l+w) \rightarrow P_{\text{new}} = \_\_ \cdot P \qquad A = lw \rightarrow A_{\text{new}} = \_\_^2 \cdot A$$

PAGE 4 — A triangle scaled

17.1 Same Rules, Any Shape

FIGURE: fig2-triangle-scaled.png (full width)

Fill in from the figure.

Original P: ______ New P: ______ Original A: ______ New A: ______

PAGE 5 — Guided practice 17.1

Work Through It

1. Give the original $P$ and $A$ of the rectangle, and the scale factor. ______
2. Give the new $P$ and $A$; confirm the ratios. ______
3. Give the triangle's original and new perimeter. ______
4. Give the triangle's original and new area; why $k^2$? ______
5. Why does perimeter only ever need one factor of $k$? ______
6. Why does area always need exactly two factors of $k$? ______

PAGE 6 — Scale it: rectangles and triangles

17.1 Find the New P and A

Find the new perimeter and area after scaling by $k$.

7. $l=5, w=2$, $k=3$ ______
8. $l=6, w=4$, $k=2$ ______
9. $l=8, w=5$, $k=4$ ______
10. square side $3$, $k=5$ ______
11. right triangle legs $6, 8$ (hyp $10$), $k=2$ ______
12. right triangle legs $5, 12$ (hyp $13$), $k=3$ ______
13. $l=10, w=6$, $k=\frac{1}{2}$ ______
14. $l=12, w=8$, $k=\frac{3}{4}$ ______

PAGE 7 — Scale it: circles

17.1 Find the New C and A

Find the new circumference and area, in terms of $\pi$.

15. $r=3$, $k=4$ ______
16. $r=5$, $k=2$ ______
17. $r=8$, $k=\frac{1}{2}$ ______

PAGE 8 — Find k

17.1 Work Backward to the Scale Factor

Find $k$.

18. rectangle $4\times6 \rightarrow 12\times18$ ______

19. rectangle $10\times15 \rightarrow 4\times6$ ______

20. legs $5,12 \rightarrow$ legs $20,48$ ______

21. circle radius $6 \rightarrow$ radius $21$ ______

22. Application. A garden plot $8$ m by $5$ m has every dimension tripled. Find the new perimeter and area.

23. Error analysis. A student says tripling a rectangle's dimensions triples its area. Using $l=4,w=2 \rightarrow l=12,w=6$, show the error.

24. Reasoning. Using $A=lw$, explain why scaling $l$ and $w$ by $k$ multiplies area by $k^2$.

Exit ticket 17.1

25. $l=9, w=4$, $k=2$. Find new $P$ and $A$. ______
26. $r=7$, $k=3$. Find new $C$ and $A$. ______

PAGE 9 — A prism scaled

17.2 One Exponent Higher

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

$$SA_{\text{new}} = k^2 \cdot SA \qquad V_{\text{new}} = k^3 \cdot V$$

PAGE 10 — A cylinder scaled

17.2 Curved Surfaces Follow Too

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

PAGE 11 — A cone scaled

17.2 Even a Derived Length Scales

FIGURE: fig5-cone-scaled.png (full width)

The slant height isn't given directly, but it's still $\times k$ — it's the hypotenuse of a triangle whose own legs both scaled.

PAGE 12 — The pattern, boxed

17.2 Every Solid, One Table

FIGURE: fig6-scale-factor-board.png (full width)

PAGE 13 — Guided practice 17.2

Work Through It

27. Identify the prism's original dimensions and $k$; give the new dimensions. ______
28. Give the original and new volume; confirm the ratio is $k^3$. ______
29. Give the original and new surface area; confirm the ratio is $k^2$. ______
30. Identify the cylinder's original $r,h$ and $k$; give the new $r,h$. ______
31. Confirm the cylinder's $V$ and $SA$ ratios. ______
32. Explain why the cone's slant height scales by $k$ too. ______

PAGE 14 — Scale it: prisms and cylinders

17.2 Find the New V and SA

Find the new volume and surface area after scaling by $k$.

33. $l=2,w=3,h=4$, $k=2$ ______
34. $l=1,w=2,h=5$, $k=3$ ______
35. cube side $3$, $k=2$ ______
36. cylinder $r=1,h=4$, $k=3$ ______
37. cylinder $r=4,h=2$, $k=\frac{1}{2}$ ______
38. cylinder $r=2,h=9$, $k=5$ ______

PAGE 15 — Scale it: cones, pyramids, spheres

17.2 Find l, V, and SA

Find the new slant height, volume, and surface area.

39. cone $r=3,h=4$, $k=2$ ______
40. cone $r=6,h=8$, $k=\frac{1}{2}$ ______
41. pyramid $b=6,h=4$, $k=2$ ______

Find the new volume and surface area.

42. sphere $r=6$, $k=3$ ______

PAGE 16 — Find k, in 3-D

17.2 Work Backward to the Scale Factor

Find $k$.

43. prism $2\times3\times4 \rightarrow 6\times9\times12$ ______

44. cylinder $r=5,h=8 \rightarrow r=1,h=1.6$ ______

45. cone $r=4,h=3 \rightarrow r=20,h=15$ ______

46. sphere $r=7 \rightarrow r=2$ ______

47. Application. A balloon radius $6$ in doubles. Find the original and new $V$ and $SA$.

48. Error analysis. A student scales a cylinder ($r=2,h=5$) by $k=3$ and claims $V$ triples. Find the real new $V$.

49. Reasoning. Why does scaling every dimension of ANY solid by $k$ multiply $SA$ by $k^2$ and $V$ by $k^3$?

Exit ticket 17.2

50. prism $3\times4\times5$, $k=2$. Find new $V,SA$. ______
51. sphere $r=12$, $k=\frac{1}{2}$. Find new $V,SA$. ______

PAGE 17 — A prism, one dimension

17.3 Volume Scales Clean. Surface Area Doesn't.

FIGURE: fig7-prism-one-dimension.png (full width)

$V=lwh$ is one product — one changed factor scales it exactly. $SA$ is a SUM of three different products — only some of them change.

PAGE 18 — A cylinder, radius only

17.3 Two Terms, Two Different Shapes

FIGURE: fig8-cylinder-radius-only.png (full width)

PAGE 19 — A cylinder, height only

17.3 Only Half the Formula Moves

FIGURE: fig9-cylinder-height-only.png (full width)

PAGE 20 — A sphere, the contrast case

17.3 One Dimension, No Choice But Uniform

FIGURE: fig10-sphere-one-dimension.png (full width)

A sphere has only one dimension — there's no second length to hold fixed. Changing it is always a uniform scaling.

PAGE 21 — Guided practice 17.3

Work Through It

52. Identify which dimension changed on the prism, and by what factor. ______
53. Give the original and new volume; why did it scale by exactly that factor? ______
54. Give the original and new surface area; why NOT that same factor? ______
55. On the radius-only cylinder, give $V$ before/after and explain the exponent. ______
56. On the height-only cylinder, give $SA$ before/after; why did only one term move? ______
57. Explain why a sphere's one-dimension change always behaves like uniform scaling. ______

PAGE 22 — One dimension: prisms

17.3 Recompute Directly

Only ONE dimension changes. Find the new V and SA directly.

58. $l=3,w=5,h=2$, only $l$ doubled ______
59. $l=4,w=2,h=6$, only $w$ tripled ______
60. $l=5,w=3,h=4$, only $h$ halved ______
61. $l=2,w=2,h=10$, only $h$ quadrupled ______
62. $l=6,w=4,h=3$, only $w$ doubled ______
63. cube $3\times3\times3$, only one edge doubled ______

PAGE 23 — One dimension: cylinders and spheres

17.3 Recompute Directly

Only the radius or only the height changes. Find the new V and SA directly.

64. $r=2,h=6$, only $r$ tripled ______
65. $r=5,h=3$, only $r$ doubled ______
66. $r=4,h=2$, only $h$ tripled ______
67. $r=3,h=10$, only $h$ halved ______
68. $r=6,h=4$, only $r$ halved ______
69. $r=1,h=8$, only $h$ quadrupled ______

Sphere — find the new V and SA.

70. $r=3$, radius doubled ______

71. $r=9$, radius tripled ______

72. Application. A glass $r=3$ cm, $h=10$ cm triples only its height. Find the original and new $V$.

73. Error analysis. A student doubles only a cylinder's radius ($r=4,h=5$) and claims $SA$ doubles. Show the real values.

74. Reasoning. Why does changing one dimension of a prism always scale $V$ by that factor, but not $SA$?

Exit ticket 17.3

75. $l=4,w=3,h=2$, only $l$ tripled. Find new $V,SA$. ______
76. $r=5,h=4$, only $r$ doubled. Find new $V,SA$. ______

PAGE 24 — Working backward: a cube

17.4 From a Volume Ratio, a Cube Root

FIGURE: fig11-cube-backward.png (full width)

$$k = \sqrt[3]{\dfrac{V_{\text{new}}}{V}}$$

PAGE 25 — Working backward: a sphere

17.4 From a Surface-Area Ratio, a Square Root

FIGURE: fig12-sphere-backward.png (full width)

$$k = \sqrt{\dfrac{SA_{\text{new}}}{SA}}$$

PAGE 26 — In context: a water tank

17.4 One Dimension, Applied

FIGURE: fig13-tank-context.png (full width)

PAGE 27 — Guided practice 17.4

Work Through It

77. Given $V$ became $27\times$, find $k$ for the cube. ______
78. Use $k$ to find how $SA$ changed. ______
79. Given $SA$ became $4\times$, find $k$ for the sphere. ______
80. Use $k$ to find how $V$ changed. ______
81. Identify the tank's changed dimension and factor; give the effect on $V$. ______
82. Which root comes from a volume ratio, and which from a surface-area ratio? ______

PAGE 28 — Find k, then the other measure

17.4 Backward, Both Ways

A solid is scaled uniformly. Find $k$, then the requested ratio.

83. $V$ becomes $8\times$. Find $k$, then the $SA$ factor. ______
84. $V$ becomes $125\times$. Find $k$, then the $SA$ factor. ______
85. $SA$ becomes $9\times$. Find $k$, then the $V$ factor. ______
86. $SA$ becomes $25\times$. Find $k$, then the $V$ factor. ______
87. $V$ becomes $\frac{1}{8}$ as large. Find $k$, then the $SA$ factor. ______
88. $SA$ becomes $\frac{1}{16}$ as large. Find $k$, then the $V$ factor. ______

PAGE 29 — Name it, then solve

17.4 Uniform, or Just One?

Name whether every dimension changed or only one, then answer.

89. A cube's edge is doubled. Find the $SA$ and $V$ factors. ______

90. A cylinder's radius is doubled, height unchanged. Find the $V$ factor. Why not $SA$ the same way? ______

91. A sphere's radius is tripled. Find the $SA$ and $V$ factors. ______

92. A prism's height is halved, length and width unchanged. Find the $V$ factor. ______

93. Application. A cube box, edge $2$ ft, is replaced by one with edge $6$ ft. Find $k$, and the cardboard and volume factors.

94. Application. A balloon's $SA$ grows from $64\pi$ to $576\pi$ ft². Find $k$ for its radius, and the volume factor.

95. Error analysis. A student is told $V$ increased $8\times$ and says $k=8$. Correct it.

96. Reasoning. Why does a volume ratio need a cube root but a surface-area ratio need a square root?

Exit ticket 17.4

97. $V$ becomes $64\times$. Find $k$, then the $SA$ factor. ______
98. A cylinder's height is tripled, radius unchanged. Find the $V$ factor. ______

PAGE 30 — Blank pairs

Your Turn

FIGURE: fig14-blank-change-frames.png (full width)

Before every problem: is every dimension changing by the same $k$, or only one? That decision is what tells you whether a shortcut power of $k$ applies, or whether you have to recompute directly.

PAGE 31 — Chapter review

Chapter 17 Review

Review 1 (G.DF.2a). A rectangle $6$ by $9$ is scaled by $k=3$.

• Give the original and new perimeter.
• Give the original and new area.
• A right triangle legs $9,12$ is scaled by the same $k=3$. Give its original and new area.
• Explain why every one of these areas scaled by $k^2$.

Review 2 (G.DF.2a). A cylindrical candle: $r=3$ cm, $h=8$ cm.

• Find its volume and surface area.
• Only its height is doubled. Find the new volume directly, and the factor.
• Find the new surface area directly; explain why it did not double.
• A second candle scales BOTH $r$ and $h$ by $k=2$. Find its new $V,SA$, and explain the difference.

Review 3 (G.DF.2 b, c). A spherical tank's $SA$ grows from $196\pi$ to $784\pi$ ft².

• Find $k$.
• Find the original and new radius.
• Find the volume factor.
• A rectangular tank has its length only tripled, width and height unchanged. Without computing, explain whether its volume increased by the same factor as the spherical tank's.