Chapter 14 — Answers

Selected answers for Chapter 14: Regular Polygons and the Circle, transcribed from the book’s appendix. They are meant for checking your work, not for copying — the book prints answers to selected exercises only.

Back to Chapter 14 · Teacher’s answers: Chapter 14 · Complete key: Answers to Exercises PDF · Printed answers: appendices PDF

Lesson 1 (pages 574–578)

3. The radii of the polygons.

5. Yes. If a triangle is equilateral, it is also equiangular.

8. No.

10. Yes. (All regular polygons are cyclic.)

13. 60°.

14. Equilateral.

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18. Three squares.

20. They are factors of it.

21. Equilateral triangles and regular pentagons.

27. Regular polygons having an even number of sides.

31. If it isn’t regular, it must not be equiangular.

33. Isosceles right triangles.

39. 22.5°.

41. 135°.

45. Apothems.

47. 30°-60° right triangles.

52. They are equal because the sides of the pentagon are chords of the circle, and equal arcs in a circle have equal chords.

54. 72°.

Lesson 2 (pages 581–584)

2. The number of sides.

4. 3.0902.

10. Radii.

14. PB bisects ∠ABC.

16. $9\sin20^\circ$.

17. 3.078.

21. The radius of the hexagon appears to be longer.

22. 108°.

26. $5\sin36^\circ\approx2.94$.

28. Approximately 2.13 angstroms.

31. 74 mm.

37. $r\sqrt{3}$.

39. $p\approx5.196r$.

41. 2.

44. By the Law of Sines.

45. Approximately 1.9.

48. Approximately 2.2.

49. 18°.

52. 3.0901699….

Lesson 3 (pages 587–590)

3. 0.

5. 1.30.

9. The perimeter of the nonagon.

14. $4a^2$.

16. $r=a\sqrt{2}$.

18. $2r^2$.

22. 628 units.

28. 31,359 square units.

31. 30°-60° right triangles.

32. $\dfrac{r}{2}$.

34. $\dfrac{\sqrt{3}}{4}r^2$.

42. They are isosceles.

43. That they are 15°.

47. All are $(2\sin15^\circ)r\approx0.518r$.

49. $\dfrac{1}{4}r^2$.

54. 7.1 cm.

57. 38 cm.

Lesson 4 (pages 593–597)

2. $\pi$.

5. The circumference of the hat.

7. Approximately 1,500,780 mi.

10. $\pi a$.

16. 704 ft/minute.

17. $14\pi$ ft.

20. 10 units.

21. 8 units.

24. AC = $\dfrac{1}{2}$AB. If a line through the center of a circle is perpendicular to a chord, it also bisects the chord.

26. 3.1415927.

28. $2\sqrt{2}$.

29. 2.83.

35. Approximately 24,880 mi.

36. Approximately 1,040 mi.

37. 1,040 mi/hour.

42. Approximately 19 mi/second.

45. $r^2=\dfrac{5}{4}x^2$.

46. $r^2=\dfrac{1}{2}y^2$.

49. $2a+2b+2c$.

52. 825 ft.

53. $\pi\approx3.14$ in.

Lesson 5 (pages 600–604)

1. 300 mi.

2. 940 mi.

3. 71,000 mi².

4. 15 mi.

5. 90 mi.

9. $x^2$.

12. 100 circular inches.

14. $3{,}600\pi$ cm².

15. $60t$ cm.

19. $36\pi$ square units.

23. $\pi^2$.

25. 6 units.

27. 3.1425.

31. $3\pi x^2$.

33. $(4-\pi)x^2$.

35. $(5\pi-8)x^2$.

36. $(3\sqrt{3}-\pi)x^2$.

42. 72.6.

43. 290.

47. $a+b$.

48. $\pi a^2+\pi ab$ or $\pi a(a+b)$.

Lesson 6 (pages 606–611)

1. A sector.

2. 45°.

3. $\dfrac{1}{4}\pi r$.

4. $\dfrac{1}{8}\pi r^2$.

5. The length of $\widehat{AB}$ in meters.

8. 15,700 m².

9. About 6,220 mi.

12. 2,009 ft².

13. 887 ft².

16. $\pi r^2$.

18. 60°.

19. $\dfrac{1}{3}\pi x$.

22. $x$.

23. About 9 in.

25. About 0.0007 second.

27. 360°.

28. $\dfrac{1}{2}\pi$.

30. 57.3°.

31. 46 in.

32. 469 in².

34. About 53°.

37. About 93 square units.

39. About 45 square units.

41. 63.66 m.

43. 103.77 m.

45. About 188 m.

47. $2\sqrt{2}$.

49. $4\pi$.

50. 8.

51. $4\pi-8$.

53. $\pi$.

58. 1,256.64 in².

62. 34.64 in.

64. 246 in².

66. $2a+b$.

67. $2a+2b$.

Review (pages 612–616)

2. It must be convex, equilateral, and equiangular.

4. 120°.

6. 48°.

9. 3.12.

10. 3.05.

12. $6.24r$.

15. About 420 turns.

18. $2\pi^2$.

20. $4\pi$.

21. $\dfrac{\pi}{8}a^2$, $\dfrac{\pi}{8}b^2$, and $\dfrac{\pi}{8}c^2$.

24. $\dfrac{1}{2}\pi r^2$.

28. $\sqrt{3}\approx1.732$.

29. $\dfrac{2}{3}\pi\approx2.094$.

31. $\dfrac{4}{3}\pi-\sqrt{3}\approx2.457$.

33. 45°.

34. 0.79.

35. $\sqrt{2}-1$.

37. 0.98.

39. About 1,037 mi.

40. About 3,034 mi.

42. About 794 mi.

45. $\sqrt{2}$ (or about 1.414).

47. 8.78.

52. $\pi R^2-\pi r^2$ or $\pi(R^2-r^2)$.

55. $4\pi$.

58. $(360-2x)^\circ$.

59. $\dfrac{180-x}{90}\pi$.

63. About 83 ft.

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