Geometry Workbook — Chapter 14: Circles: Arc Length and Sector Area

SOL G.PC.3 (a, d, e) · Companion to Textbook Chapter 14

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 112.

PAGE 1 — Chapter opener

Chapter 14 · Circles: Arc Length and Sector Area

Standard G.PC.3 (a, d, e)

In this chapter you will:

• Say what fraction of a circle a central angle opens onto, and reduce it
• Keep arc measure and arc length apart
• Calculate the length of an arc, exactly and as a decimal
• Calculate the area of a sector, exactly and as a decimal
• Work backwards from a length or an area to the angle or the radius
• Say what happens when the radius changes

Words to know: circumference · area of a circle · arc measure · arc length · sector · sector area · central angle · proportional · exact form · perimeter of a sector

Conventions: exact in terms of $\pi$ first, decimals only when asked. Reduce the fraction before using it. An arc length carries the units of the radius; a sector area carries square units.

PAGE 2 — The whole chapter on one page

One Fraction, Two Totals

There are not two formulas here. There is one fraction, $\tfrac{n}{360}$, applied to two different totals — the circumference for a length, the area for a region.

Before every problem:

• What is being measured — a distance along the edge, or a region inside?
• What is $\tfrac{n}{360}$, in lowest terms?
• Do my units match the question — plain units, or square units?

PAGE 3 — Measure is not length

14.1 Two Words, Two Quantities

FIGURE: fig1-measure-is-not-length.png (full width)

Fill in the table from the figure.

The error this chapter exists to prevent: answering "$60$" when asked for a length. $60$ is the measure.

PAGE 4 — The fraction

14.1 What Share of the Circle?

FIGURE: fig2-fraction-of-a-circle.png (full width)

A central angle of $n°$ opens onto $\dfrac{n}{360}$ of the circle.

• The arc is that fraction of the ____________________.
• The sector is that fraction of the ____________________.

FIGURE: fig3-fraction-table.png (full width)

PAGE 5 — Guided practice 14.1

Work Through It

1. Give the arc measure in each circle. ______
2. Give the arc length in each circle, and say why they differ. ______
3. Say what would happen to both if the radius were tripled. ______
4. Give the central angle and the fraction it opens onto. ______
5. Say what that fraction is a fraction of, for an arc and for a sector. ______
6. Give the reduced fraction for $72°$ and say what it means in words. ______

PAGE 6 — Fractions

14.1 Reduce First

Give each fraction in lowest terms.

7. $60°$ ______
8. $90°$ ______
9. $120°$ ______
10. $45°$ ______
11. $30°$ ______
12. $180°$ ______
13. $240°$ ______
14. $270°$ ______
15. $80°$ ______
16. $100°$ ______

Give the central angle.

17. A sector is $\tfrac14$ of its circle. ______
18. A sector is $\tfrac13$ of its circle. ______
19. A sector is $\tfrac25$ of its circle. ______
20. A sector is $\tfrac56$ of its circle. ______

PAGE 7 — Naming, comparing, reasoning

14.1 Words and Why

21. Name the region bounded by two radii and the arc between them. ______

22. Name the curved edge of that region on its own. ______

23. Circle $A$ has radius $2$ and circle $B$ has radius $18$. Each has a $70°$ arc. Compare the two arc measures. ______

24. Application. A circular running track is marked into $8$ equal sections by radial lines. Give the fraction of the track in one section, and its central angle.

25. Error analysis. A student is asked for the length of a $50°$ arc and answers "$50$." Say what they gave instead, and what else they needed to know.

26. Reasoning. Explain why two arcs can have equal measure and unequal length, but two arcs in the same circle with equal measure must have equal length.

Exit ticket 14.1

27. Give the fraction of a circle opened onto by a $150°$ central angle, in lowest terms. ______
28. State, in one sentence each, what arc measure tells you and what arc length tells you.

PAGE 8 — Where arc length comes from

14.2 Cutting Down the Circumference

FIGURE: fig4-deriving-arc-length.png (full width)

$$C = 2\pi r \qquad L = \frac{n}{360} \times 2\pi r$$

FIGURE: fig5-arc-length-worked.png (full width)

Copy the four steps of the worked example, in order.

PAGE 9 — Length grows with the radius

14.2 Same Angle, Four Circles

FIGURE: fig6-length-scales-with-radius.png (full width)

Complete from the figure.

Every row is $\tfrac16$ of its own circumference. Arc length is proportional to the radius.

PAGE 10 — Guided practice 14.2

Work Through It

29. Give the circumference of the whole circle. ______
30. Give the fraction and the arc length, and say how they are connected. ______
31. Give the units of the arc length, and say why. ______
32. Say what the number $40$ measures, and in what units. ______
33. Give the arc length when the radius is $9$. ______
34. Say what happens to the arc length when the radius is doubled. ______

PAGE 11 — Arc length in terms of pi

14.2 Exact Answers

Give the arc length in terms of $\pi$.

35. $r = 10$, $n = 90°$ ______
36. $r = 6$, $n = 120°$ ______
37. $r = 15$, $n = 72°$ ______
38. $r = 8$, $n = 45°$ ______
39. $r = 24$, $n = 30°$ ______
40. $r = 20$, $n = 54°$ ______
41. $r = 21$, $n = 120°$ ______
42. $r = 5$, $n = 216°$ ______

Give the arc length to the nearest hundredth.

43. $r = 9$, $n = 40°$ ______
44. $r = 12$, $n = 60°$ ______
45. $r = 10$, $n = 90°$ ______

PAGE 12 — Working backwards

14.2 Solve for What Is Missing

FIGURE: fig10-working-backwards.png (full width)

Find the missing quantity.

46. Radius $12$, arc length $4\pi$. Give the central angle. ______
47. Radius $10$, arc length $5\pi$. Give the central angle. ______
48. Radius $24$, arc length $4\pi$. Give the central angle. ______
49. A $60°$ arc has length $4\pi$. Give the radius. ______
50. A $120°$ arc has length $4\pi$. Give the radius. ______
51. A $72°$ arc has length $6\pi$. Give the radius. ______

PAGE 13 — Apply and reason 14.2

14.2 In Context

52. Application. A circular garden path has radius $30$ feet. A section is bounded by a $144°$ central angle. Give its length in terms of $\pi$, then to the nearest hundredth.

53. Error analysis. A student calculates a $90°$ arc in a circle of radius $8$ as $\tfrac{90}{360} \times \pi(8)^2 = 16\pi$. Identify the mistake and give the correct arc length.

54. Reasoning. Two arcs have the same length. Explain why their measures need not be equal, and give a pair of examples that shows it.

Exit ticket 14.2

55. Give the length of a $120°$ arc in a circle of radius $6$, in terms of $\pi$. ______
56. A $30°$ arc has length $4\pi$. Give the radius. ______

PAGE 14 — Where sector area comes from

14.3 Cutting Down the Area

FIGURE: fig7-deriving-sector-area.png (full width)

$$A = \pi r^2 \qquad A_{\text{sector}} = \frac{n}{360} \times \pi r^2$$

FIGURE: fig8-sector-area-worked.png (full width)

Square units. The formula squares the radius, and the units square with it.

The check that never fails. Put $n = 360$ in and you must get $\pi r^2$ back. So a sector area can never be bigger than ____________________.

PAGE 15 — Guided practice 14.3

Work Through It

57. Give the area of the whole circle. ______
58. Give the fraction and the sector area, and say how they are connected. ______
59. Give the units of the sector area, and say why they are squared. ______
60. Say which quantity gets squared in the formula and which does not. ______
61. State the check that a sector area can never fail. ______
62. Say what is different between the two formulas, and what is the same. ______

PAGE 16 — Sector area in terms of pi

14.3 Exact Answers

Give the sector area in terms of $\pi$.

63. $r = 12$, $n = 60°$ ______
64. $r = 10$, $n = 90°$ ______
65. $r = 9$, $n = 40°$ ______
66. $r = 6$, $n = 120°$ ______
67. $r = 15$, $n = 72°$ ______
68. $r = 8$, $n = 45°$ ______
69. $r = 24$, $n = 30°$ ______
70. $r = 20$, $n = 54°$ ______
71. $r = 14$, $n = 180°$ ______
72. $r = 21$, $n = 120°$ ______

Give the sector area to the nearest hundredth.

73. $r = 10$, $n = 90°$ ______
74. $r = 6$, $n = 120°$ ______

PAGE 17 — Backwards, apply, reason 14.3

14.3 Solve and Explain

Find the missing quantity.

75. Radius $12$, sector area $24\pi$. Give the central angle. ______

76. Radius $9$, sector area $9\pi$. Give the central angle. ______

77. Radius $24$, sector area $48\pi$. Give the central angle. ______

78. A $60°$ sector has area $24\pi$. Give the radius. ______

79. A $120°$ sector has area $48\pi$. Give the radius. ______

80. Application. A lawn sprinkler sprays $15$ feet and sweeps $72°$. Give the area it waters in terms of $\pi$, then to the nearest hundredth.

81. Error analysis. A student finds a $45°$ sector of a circle of radius $8$ as $\tfrac{45}{360} \times 2\pi(8) = 2\pi$. Identify the mistake, give the correct area, and say which unit would have caught it.

82. Reasoning. A sector's central angle is doubled and its radius is left alone. Say what happens to its area, and explain why the answer is not "it is squared."

Exit ticket 14.3

83. Give the area of a $45°$ sector of a circle of radius $8$, in terms of $\pi$. ______
84. A $90°$ sector has area $25\pi$. Give the radius. ______

PAGE 18 — One figure, two answers

14.4 Which Does the Question Want?

FIGURE: fig9-length-and-area-together.png (full width)

FIGURE: fig12-formula-board.png (full width)

PAGE 19 — Changing the radius

14.4 One Factor of r, or Two

FIGURE: fig11-doubling-the-radius.png (full width)

A length carries one factor of $r$. An area carries two.

PAGE 20 — Guided practice 14.4

Work Through It

85. Give the arc length and the sector area, with units. ______
86. Say which of the two a question about fencing would want. ______
87. Say what tells you which row of the formula board you are on. ______
88. Give the equation you would write to find the radius. ______
89. Give the factor for the arc and the factor for the area. ______
90. Say which of the three context quantities is an area, and how you know. ______

PAGE 21 — Name it, then find it

14.4 Arc or Sector?

Name the quantity — arc length or sector area — and give it in terms of $\pi$.

91. The curved edge of a $90°$ sector of radius $12$. ______
92. The region inside a $90°$ sector of radius $12$. ______
93. The curved edge of a $30°$ sector of radius $24$. ______
94. The region inside a $30°$ sector of radius $24$. ______
95. The curved edge of a $72°$ sector of radius $15$. ______
96. The region inside a $72°$ sector of radius $15$. ______

Give both.

97. A $60°$ sector of radius $12$. ______
98. A $120°$ sector of radius $6$. ______

Give the perimeter.

99. A $60°$ sector of radius $12$. ______
100. A $90°$ sector of radius $10$. ______

PAGE 22 — Scaling

14.4 What Changes, and By How Much

101. Radius doubled, angle unchanged. Factor for the arc length. ______
102. Radius doubled, angle unchanged. Factor for the area. ______
103. Angle doubled, radius unchanged. Factor for the arc length and for the area. ______
104. Radius halved, angle unchanged. Factor for the area. ______

FIGURE: fig13-arcs-and-sectors-in-context.png (full width)

PAGE 23 — In context

14.4 Crust, Clock, Sprinkler

105. Application. A pizza of radius $8$ in is cut into $8$ equal slices. Give the crust length of one slice and the area of one slice, both in terms of $\pi$.

106. Application. A minute hand is $12$ cm long. Give the distance its tip travels in $20$ minutes, in terms of $\pi$.

107. Application. A sprinkler reaches $15$ ft and sweeps $72°$. Give the watered area in terms of $\pi$, then to the nearest hundredth.

108. Error analysis. A student is asked for the perimeter of a $90°$ sector of radius $10$ and answers $5\pi$. Say what they gave, and give the perimeter.

109. Reasoning. Explain why the units alone are enough to tell you whether an answer belongs to arc length or to sector area.

110. Reasoning. Explain why doubling the radius does not double the sector area, using the formula rather than the picture.

Exit ticket 14.4

111. A $120°$ sector has radius $21$. Give its arc length and its sector area, in terms of $\pi$. ______
112. Give the one question to ask before choosing between the two formulas. ______

PAGE 24 — Blank sectors

Your Turn

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

For every problem in this chapter:

• Write $\dfrac{n}{360}$ in lowest terms before either formula.
• Ask what is being measured — the curved edge, or the region inside.
• Write the total you are taking a fraction of: $2\pi r$ or $\pi r^2$.
• Check the units. Plain units for a length, square units for an area.
• Give the answer in terms of $\pi$ unless a decimal was asked for.

PAGE 25 — Chapter review

Chapter 14 Review

Review 1 (G.PC.3a). A circle has radius $18$, and a central angle of $100°$ is drawn.

• Give the fraction of the circle that angle opens onto, in lowest terms.
• Give what that fraction is a fraction of, for the arc and for the sector.
• Give the arc length and the sector area, in terms of $\pi$.
• Say which answers change if the radius becomes $36$ with the angle unchanged, and by what factor each changes.

Review 2 (G.PC.3d). A circle of radius $20$ has an arc of length $6\pi$.

• Give the central angle.
• Give the arc's measure and its length, and say which units each carries.
• A second circle of radius $40$ has an arc of the same measure. Give its length.
• Say what stayed the same between the two circles and what did not, and why.

Review 3 (G.PC.3 d, e). A $120°$ sector is cut from a circle of radius $6$.

• Give the arc length and the sector area, in terms of $\pi$.
• Give the perimeter of the sector, and say why it is not the arc length.
• Give the sector area as a decimal to the nearest hundredth.
• Check your sector area against the whole circle's area, and say what that check rules out.