Appendix A — Answer Key, Chapter 14: Circles: Arc Length and Sector Area

SOL G.PC.3 (a, d, e) · Covers textbook Chapter 14 and the companion workbook. Item numbers match the textbook; workbook items are the same problems, so this key serves both. Item numbers run continuously from 1 to 112 across the chapter.

Conventions used in every answer below. A central angle of $n°$ opens onto $\tfrac{n}{360}$ of the circle, and that one fraction is applied to the circumference for an arc length and to the area for a sector area. Fractions are reduced before they are used. Answers are exact in terms of $\pi$ first; a decimal is given only where the item asks for one, rounded to the nearest hundredth and written as about. An arc length carries the units of the radius; a sector area carries square units; an arc measure carries degrees and is not a length at all.

Before either formula, ask what is being measured — a distance along the curved edge, or a region inside. Almost every wrong answer below would come from answering the other question.

Lesson 14.1 — The Fraction of a Circle

Guided practice

1. $60°$ in both. Arc measure does not depend on the size of the circle.
2. $\pi$ in the small circle and $2\pi$ in the large one. They differ because arc length is a distance, and the larger circle's radius is twice as long, so the same share of it is twice as far.
3. The measure would stay $60°$. The length would be tripled.
4. $90°$, opening onto $\tfrac{90}{360} = \tfrac14$ of the circle.
5. It is a fraction of the circumference when you want the arc, and of the area when you want the sector.
6. $\tfrac{72}{360} = \tfrac15$ — a $72°$ sector is one fifth of its circle.

Independent practice

7. $\tfrac{60}{360} =$ $\tfrac16$
8. $\tfrac{90}{360} =$ $\tfrac14$
9. $\tfrac{120}{360} =$ $\tfrac13$
10. $\tfrac{45}{360} =$ $\tfrac18$
11. $\tfrac{30}{360} =$ $\tfrac{1}{12}$
12. $\tfrac{180}{360} =$ $\tfrac12$
13. $\tfrac{240}{360} =$ $\tfrac23$
14. $\tfrac{270}{360} =$ $\tfrac34$
15. $\tfrac{80}{360} =$ $\tfrac29$
16. $\tfrac{100}{360} =$ $\tfrac{5}{18}$
17. $\tfrac14 (360) =$ $90°$
18. $\tfrac13 (360) =$ $120°$
19. $\tfrac25 (360) =$ $144°$
20. $\tfrac56 (360) =$ $300°$
21. A sector.
22. An arc.
23. Equal — both $70°$. Measure is a share of a full turn, so it is the same in every circle. Their lengths are not equal: circle $B$'s arc is nine times as long.
24. $\tfrac18$ of the track, and a central angle of $360 \div 8 =$ $45°$.
25. They gave the arc's measure, in degrees, when the question asked for its length, in units of distance. To answer the question asked they also needed the radius — measure alone determines no length, because the same $50°$ arc is a different length in every different circle.
26. Two arcs of equal measure in different circles are the same share of two different circumferences, and equal shares of unequal totals are unequal. Within one circle the total is the same for both, so equal shares of it must be equal.

Exit ticket 14.1

27. $\tfrac{150}{360} =$ $\tfrac{5}{12}$
28. Arc measure tells you what share of a full turn the arc covers, in degrees, and is the same in every circle. Arc length tells you how far it is to travel along the arc, in the units of the radius, and grows with the circle.

Lesson 14.2 — Arc Length

Guided practice

29. $C = 2\pi(9) =$ $18\pi$
30. $\tfrac{40}{360} = \tfrac19$, and the arc is $\tfrac19$ of $18\pi$, which is $2\pi$. The arc length is that fraction of the circumference — the formula is nothing more than that sentence.
31. Centimetres — the same units as the radius, because an arc length is a distance.
32. $40$ measures the arc's (and the central angle's) measure, in degrees. It is not a number of centimetres.
33. $3\pi$
34. It doubles. Arc length is proportional to the radius.

Independent practice

35. $\tfrac14 (20\pi) =$ $5\pi$
36. $\tfrac13 (12\pi) =$ $4\pi$
37. $\tfrac15 (30\pi) =$ $6\pi$
38. $\tfrac18 (16\pi) =$ $2\pi$
39. $\tfrac{1}{12} (48\pi) =$ $4\pi$
40. $\tfrac{3}{20} (40\pi) =$ $6\pi$
41. $\tfrac13 (42\pi) =$ $14\pi$
42. $\tfrac35 (10\pi) =$ $6\pi$
43. $\tfrac19 (18\pi) = 2\pi \approx$ $6.28$
44. $\tfrac16 (24\pi) = 4\pi \approx$ $12.57$
45. $\tfrac14 (20\pi) = 5\pi \approx$ $15.71$
46. $\tfrac{n}{360}(24\pi) = 4\pi \Rightarrow \tfrac{n}{360} = \tfrac16 \Rightarrow n =$ $60°$
47. $\tfrac{n}{360}(20\pi) = 5\pi \Rightarrow \tfrac{n}{360} = \tfrac14 \Rightarrow n =$ $90°$
48. $\tfrac{n}{360}(48\pi) = 4\pi \Rightarrow \tfrac{n}{360} = \tfrac{1}{12} \Rightarrow n =$ $30°$
49. $\tfrac16 (2\pi r) = 4\pi \Rightarrow \tfrac{\pi r}{3} = 4\pi \Rightarrow r =$ $12$
50. $\tfrac13 (2\pi r) = 4\pi \Rightarrow \tfrac{2\pi r}{3} = 4\pi \Rightarrow r =$ $6$
51. $\tfrac15 (2\pi r) = 6\pi \Rightarrow \tfrac{2\pi r}{5} = 6\pi \Rightarrow r =$ $15$
52. $\tfrac{144}{360} = \tfrac25$ and $C = 60\pi$, so the section is $\tfrac25 (60\pi) =$ $24\pi$ feet, or about $75.40$ feet.
53. They used the area formula, $\pi r^2$, inside the arc length calculation. An arc is a fraction of the circumference, $2\pi r$. Correct: $\tfrac14 (16\pi) =$ $4\pi$. The units give it away — $16\pi$ would be square units, and a length cannot be square units.
54. Equal lengths do not force equal measures, because the length depends on the radius as well as the measure. A small share of a big circle can be exactly as long as a big share of a small circle. Example: a $60°$ arc of radius $12$ has length $4\pi$, and so does a $120°$ arc of radius $6$ — same length, measures differing by a factor of two.

Exit ticket 14.2

55. $\tfrac13 (12\pi) =$ $4\pi$
56. $\tfrac{1}{12} (2\pi r) = 4\pi \Rightarrow \tfrac{\pi r}{6} = 4\pi \Rightarrow r =$ $24$

Lesson 14.3 — Sector Area

Guided practice

57. $A = \pi(12)^2 =$ $144\pi$
58. $\tfrac{60}{360} = \tfrac16$, and the sector is $\tfrac16$ of $144\pi$, which is $24\pi$. Same fraction as the arc; the total is the area instead of the circumference.
59. Square inches. The formula squares the radius, and squaring a length gives an area.
60. The radius is squared. The fraction $\tfrac{n}{360}$ is not — it is a share, and a share of an area is still that same share.
61. Put $n = 360$ in and you must get the whole circle back: $\tfrac{360}{360}\pi r^2 = \pi r^2$. So a sector area can never exceed $\pi r^2$, and any answer that does is wrong without needing the right one for comparison.
62. The same: the fraction $\tfrac{n}{360}$, computed identically. Different: the total it multiplies — $2\pi r$ for an arc, $\pi r^2$ for a sector — and therefore the units, linear against square.

Independent practice

63. $\tfrac16 (144\pi) =$ $24\pi$
64. $\tfrac14 (100\pi) =$ $25\pi$
65. $\tfrac19 (81\pi) =$ $9\pi$
66. $\tfrac13 (36\pi) =$ $12\pi$
67. $\tfrac15 (225\pi) =$ $45\pi$
68. $\tfrac18 (64\pi) =$ $8\pi$
69. $\tfrac{1}{12} (576\pi) =$ $48\pi$
70. $\tfrac{3}{20} (400\pi) =$ $60\pi$
71. $\tfrac12 (196\pi) =$ $98\pi$
72. $\tfrac13 (441\pi) =$ $147\pi$
73. $\tfrac14 (100\pi) = 25\pi \approx$ $78.54$
74. $\tfrac13 (36\pi) = 12\pi \approx$ $37.70$
75. $\pi(12)^2 = 144\pi$, and $\tfrac{24\pi}{144\pi} = \tfrac16$, so $n = \tfrac16 (360) =$ $60°$
76. $\pi(9)^2 = 81\pi$, and $\tfrac{9\pi}{81\pi} = \tfrac19$, so $n = \tfrac19 (360) =$ $40°$
77. $\pi(24)^2 = 576\pi$, and $\tfrac{48\pi}{576\pi} = \tfrac{1}{12}$, so $n = \tfrac{1}{12} (360) =$ $30°$
78. $\tfrac16 \pi r^2 = 24\pi \Rightarrow \pi r^2 = 144\pi \Rightarrow r^2 = 144 \Rightarrow r =$ $12$
79. $\tfrac13 \pi r^2 = 48\pi \Rightarrow \pi r^2 = 144\pi \Rightarrow r^2 = 144 \Rightarrow r =$ $12$
80. $\tfrac{72}{360} = \tfrac15$ and $A = 225\pi$, so the watered region is $\tfrac15 (225\pi) =$ $45\pi$ square feet, or about $141.37$ square feet.
81. They used the circumference, $2\pi r$, inside the sector area calculation. A sector is a fraction of the area, $\pi r^2$. Correct: $\tfrac18 (64\pi) =$ $8\pi$. Square inches would have caught it — their answer came out in plain inches, which cannot measure a region.
82. The area doubles. The formula multiplies the whole circle's area by $\tfrac{n}{360}$, and $n$ appears there to the first power — doubling $n$ doubles the share, nothing more. Squaring happens to the radius, which was left alone.

Exit ticket 14.3

83. $\tfrac18 (64\pi) =$ $8\pi$
84. $\tfrac14 \pi r^2 = 25\pi \Rightarrow \pi r^2 = 100\pi \Rightarrow r^2 = 100 \Rightarrow r =$ $10$

Lesson 14.4 — Putting It Together

Guided practice

85. Arc $5\pi$ metres; sector $25\pi$ square metres.
86. The arc length, $5\pi$ m — fencing runs along the curved edge, so it is a distance.
87. The words of the question, not its numbers: whether it asks about a distance along the edge or a region inside. The units of your answer are then the check.
88. $\tfrac{40}{360}(2\pi r) = 2\pi$, which gives $r = 9$.
89. Arc $\times 2$; area $\times 4$.
90. The sprinkler's $45\pi$ ft² — it is the only one in square units, and it measures a region rather than a distance travelled.

Independent practice

91. Arc length. $\tfrac14 (24\pi) =$ $6\pi$
92. Sector area. $\tfrac14 (144\pi) =$ $36\pi$
93. Arc length. $\tfrac{1}{12} (48\pi) =$ $4\pi$
94. Sector area. $\tfrac{1}{12} (576\pi) =$ $48\pi$
95. Arc length. $\tfrac15 (30\pi) =$ $6\pi$
96. Sector area. $\tfrac15 (225\pi) =$ $45\pi$
97. Arc $\tfrac16 (24\pi) =$ $4\pi$; sector $\tfrac16 (144\pi) =$ $24\pi$
98. Arc $\tfrac13 (12\pi) =$ $4\pi$; sector $\tfrac13 (36\pi) =$ $12\pi$
99. Two radii plus the arc: $12 + 12 + 4\pi =$ $24 + 4\pi$
100. Two radii plus the arc: $10 + 10 + 5\pi =$ $20 + 5\pi$
101. $\times 2$
102. $\times 4$
103. Arc $\times 2$ and area $\times 2$. The angle enters both formulas to the first power, so doubling it doubles both.
104. $\times \tfrac14$
105. One slice is $360 \div 8 = 45°$. Crust $\tfrac18 (16\pi) =$ $2\pi$ inches; slice $\tfrac18 (64\pi) =$ $8\pi$ square inches.
106. $20$ minutes is $20 \times 6 = 120°$, so the tip travels $\tfrac13 (24\pi) =$ $8\pi$ centimetres.
107. $\tfrac15 (225\pi) =$ $45\pi$ square feet, or about $141.37$ square feet.
108. They gave the arc length — the curved edge alone. A sector's boundary is the arc and the two radii, so the perimeter is $10 + 10 + 5\pi =$ $20 + 5\pi$.
109. Because the two formulas produce different kinds of quantity and cannot be confused once the units are attached: an arc length multiplies a length by a pure fraction and stays linear, while a sector area starts from $\pi r^2$ and is square. So metres means arc, square metres means sector, whatever the working looked like.
110. Because $r$ appears squared in $\tfrac{n}{360}\pi r^2$. Replacing $r$ by $2r$ gives $\tfrac{n}{360}\pi (2r)^2 = \tfrac{n}{360}\pi \cdot 4r^2$, which is four times the original, not two. In the arc formula $r$ appears to the first power, so there the factor really is two.

Exit ticket 14.4

111. Arc $\tfrac13 (42\pi) =$ $14\pi$; sector $\tfrac13 (441\pi) =$ $147\pi$
112. "Is the question asking for a distance along the curved edge, or for the region inside?" — arc length for the first, sector area for the second.

Chapter 14 Review — answers

Review 1 (G.PC.3a).

• $\tfrac{100}{360} =$ $\tfrac{5}{18}$
• Of the circumference $2\pi(18) = 36\pi$ for the arc, and of the area $\pi(18)^2 = 324\pi$ for the sector.
• Arc $\tfrac{5}{18}(36\pi) =$ $10\pi$; sector $\tfrac{5}{18}(324\pi) =$ $90\pi$
• Both change, by different factors. Doubling the radius to $36$ leaves the fraction alone but doubles the circumference and quadruples the area, so the arc becomes $20\pi$ — a factor of $2$ — and the sector becomes $360\pi$ — a factor of $4$. The arc measure, $100°$, is the thing that does not change.

Review 2 (G.PC.3d).

• $\tfrac{n}{360}(40\pi) = 6\pi \Rightarrow \tfrac{n}{360} = \tfrac{3}{20} \Rightarrow n =$ $54°$
• Measure $54°$, in degrees; length $6\pi$, in the units of the radius. Two different quantities describing one arc.
• Same measure, $54°$, in a circle of radius $40$: $\tfrac{3}{20}(80\pi) =$ $12\pi$ — twice as long.
• What stayed the same is the measure, $54°$, and therefore the fraction $\tfrac{3}{20}$ — both are shares of a full turn and take no notice of size. What changed is the length, because the circumference it is a share of doubled when the radius doubled.

Review 3 (G.PC.3 d, e).

• Arc $\tfrac13 (12\pi) =$ $4\pi$; sector $\tfrac13 (36\pi) =$ $12\pi$
• Perimeter $6 + 6 + 4\pi =$ $12 + 4\pi$. It is not the arc length because the sector is bounded by two radii as well as the arc; the arc is only the curved part of the boundary.
• $12\pi \approx$ $37.70$
• The whole circle has area $\pi(6)^2 = 36\pi$, and $12\pi$ is exactly $\tfrac13$ of it — consistent with a $120°$ angle. The check rules out any answer larger than $36\pi$, and it does so without knowing the correct value, which is what makes it worth running every time.

Every item in Chapter 14 is answered above: 1 to 112, plus the three chapter reviews.