Grade 8 Workbook — Chapter 12: Area and Perimeter of Composite Figures

SOL 8.MG.5 · 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/. Every numbered item is the same problem as in the textbook, so one answer key serves both. Item numbers run continuously from 1 to 104.

PAGE 1 — Chapter opener

Chapter 12 · Area and Perimeter of Composite Figures

Standard 8.MG.5

In this chapter you will:

• Cut a figure into rectangles, squares, triangles, trapezoids, parallelograms, circles, and semicircles, and add up the pieces to get the area
• Find the same area a second way, and use the agreement as a check
• Find the area of a figure with a piece removed
• Trace a boundary to find a perimeter — and see why adding the pieces' perimeters is wrong
• Use the arc of a semicircle, but never its diameter, in a perimeter
• Solve real problems about tile, fence, trim, turf, glass, and cost

Words to know: composite figure · subdivide (decompose) · boundary · interior edge · subdivision line · semicircle · diameter · arc · circumference · perimeter · area · exact value · approximation

Pi: use $\pi \approx 3.14$. Give the exact answer in terms of $\pi$ first, then the approximation.

PAGE 2 — Cutting a figure into pieces

12.1 Subdividing a Figure

FIGURE: fig18-decomposition-menu.png (full width)

The pieces you are allowed to use, and their areas. Fill in each formula.

FIGURE: fig1-l-shape-two-ways.png (full width)

1. Cut across. $A = (10)(3) + (6)(3) = \underline{\hspace{1.5cm}} + \underline{\hspace{1.5cm}} = \underline{\hspace{2cm}}$ cm$^2$

2. Cut down. $A = (6)(6) + (4)(3) = \underline{\hspace{1.5cm}} + \underline{\hspace{1.5cm}} = \underline{\hspace{2cm}}$ cm$^2$

Why did both give the same number? _______________________________________________

PAGE 3 — Subtraction, and pieces that are not rectangles

Three More Ways to See a Figure

FIGURE: fig2-subtraction-method.png (half width)

3. By subtraction. $A = (10)(6) - (4)(3) = \underline{\hspace{2cm}}$ cm$^2$

FIGURE: fig3-house-rectangle-triangle.png (half width)

4. Pieces: ____________ and ____________

   Rectangle: $\underline{\hspace{1cm}} \times \underline{\hspace{1cm}} = \underline{\hspace{1.5cm}}$ Triangle: $\tfrac{1}{2}(\underline{\hspace{1cm}})(\underline{\hspace{1cm}}) = \underline{\hspace{1.5cm}}$

   Total area: _______________

Careful: the triangle's height is not the height of the whole figure, and the slanted sides are not heights.

FIGURE: fig4-trapezoid-on-rectangle.png (half width)

5. Rectangle: _______________ Trapezoid: $\tfrac{1}{2}(\underline{\hspace{1cm}} + \underline{\hspace{1cm}})(\underline{\hspace{1cm}}) = \underline{\hspace{1.5cm}}$

   Total area: _______________

FIGURE: fig5-parallelogram-on-rectangle.png (half width)

6. Parallelogram base: ______ height: ______ area: ______

   Total area: _______________

PAGE 4 — Practice: find the area

Area Practice

FIGURE: fig13-practice-area-set.png (full width)

7. Figure 13a. Find the area two different ways.

   Way 1: _______________________________________________

   Way 2: _______________________________________________

   Do they agree? ______

8. Figure 13b. Pieces used: ____________ and ____________ Area: _______________

9. Figure 13c. With the trapezoid formula: _______________

   As a rectangle plus two triangles: _______________

10. Figure 13d. Vertical strips: _______________ Horizontal strips: _______________

PAGE 5 — Explain and repair

Thinking About Subdivisions

11. Explain. Why must cutting Figure 1 across and cutting it down give the same area?

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12. Explain. When is subtraction faster than adding pieces? Name a figure from this lesson where adding is better.

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13. Find the error. For Figure 3 a student wrote $\tfrac{1}{2}(12)(9) = 54$ for the triangle.

    What went wrong? _______________________________________________

    Correct total area: _______________

14. Find the error. For Figure 4 a student wrote $\tfrac{1}{2}(14 + 6)(11) = 110$ for the trapezoid.

    What went wrong? _______________________________________________

    Correct total area: _______________

15. Apply it. A reading nook has the shape of Figure 13a, measured in feet. Tile costs $6 per square foot.

    Area: ____________ ft$^2$ Cost: $____________

16. Explain. A classmate says a dashed subdivision line "adds a little area where the pieces meet." Why is that wrong?

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PAGE 6 — Exit ticket 12.1

Exit Ticket · Lesson 12.1

Name: ________________________ Date: ____________

17. Area of the figure in Figure 3: _______________

18. Area of the figure in Figure 4: _______________

19. The two subdivisions of Figure 1, and the total each gives:

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20. What does it mean to subdivide a plane figure, and why do we do it?

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PAGE 7 — Curved pieces

12.2 Circles and Semicircles

Complete. $A_{\text{circle}} = \underline{\hspace{2cm}}$ and $A_{\text{semicircle}} = \underline{\hspace{2cm}}$

The straight edge of a semicircle is the ____________. The curved edge is the ____________.

FIGURE: fig6-semicircle-on-rectangle.png (half width)

The rectangle is $10$ cm wide, so the semicircle's diameter is ______ cm and its radius is ______ cm.

21. Area of a circle with $r = 4$ cm: exact ____________ approximate ____________

22. Area of a semicircle with $r = 6$ cm: exact ____________ approximate ____________

23. Figure 6. Rectangle ____________ Semicircle ____________ Total ____________

PAGE 8 — Taking a piece away

Area with a Piece Removed

The rule: area of the whole $-$ area of the removed piece.

FIGURE: fig7-circle-removed.png (half width)

24. Grass area: $(20)(12) - \pi(\underline{\hspace{0.8cm}})^2 = \underline{\hspace{2.5cm}} \approx \underline{\hspace{2.5cm}}$ m$^2$

FIGURE: fig9-semicircle-bite.png (half width)

25. Area: $(12)(8) - \tfrac{1}{2}\pi(\underline{\hspace{0.8cm}})^2 = \underline{\hspace{2.5cm}} \approx \underline{\hspace{2.5cm}}$ cm$^2$

26. The rectangle in Figure 6 is $10$ cm wide. Why is the radius $5$ cm and not $10$ cm?

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PAGE 9 — Practice: area with curves

Curved-Piece Practice

FIGURE: fig14-practice-circle-set.png (full width)

Give each answer exactly in terms of $\pi$, then approximately with $\pi \approx 3.14$.

31. Explain. Find the area of Figure 14c a second way by joining the two semicircles into one circle.

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    Do the two methods agree? ______

PAGE 10 — Errors and applications

Watch the Radius

32. Find the error. For Figure 14a a student wrote $\tfrac{1}{2}\pi(6)^2 = 18\pi$.

    What went wrong? _______________________________________________

    Correct total area: _______________

33. Find the error. For Figure 14d a student wrote $216 + 9\pi$.

    What went wrong? _______________________________________________

    Correct area: _______________

34. Apply it. Glass for the window in Figure 11.

    Exact: ____________ To the nearest hundredth: ____________ ft$^2$

35. Apply it. The countertop in Figure 14b is cut from a $12$ in by $12$ in slab. Material costs $0.40 per square inch.

    Area: ____________ in$^2$ Cost: $____________

36. Explain. Why is $60 + 12.5\pi$ a more accurate answer than $99.25$?

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PAGE 11 — Exit ticket 12.2

Exit Ticket · Lesson 12.2

Name: ________________________ Date: ____________

37. Area of the figure in Figure 9: exact ____________ approximate ____________

38. Area of the figure in Figure 6: exact ____________ approximate ____________

39. Area of the figure in Figure 14d: exact ____________ approximate ____________

40. For area, what is the difference between attaching a semicircle and removing one?

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PAGE 12 — The big idea of perimeter

12.3 Perimeter Is the Boundary

The perimeter of a composite figure is NOT the sum of the perimeters of its pieces.

FIGURE: fig8-interior-edges.png (full width)

41. Apart: $2(12 + 4) = \underline{\hspace{1.5cm}}$ and $2(4 + 6) = \underline{\hspace{1.5cm}}$, total $\underline{\hspace{1.5cm}}$ units of edge.

42. Joined: trace the boundary.

    $\underline{\hspace{0.8cm}} + \underline{\hspace{0.8cm}} + \underline{\hspace{0.8cm}} + \underline{\hspace{0.8cm}} + \underline{\hspace{0.8cm}} + \underline{\hspace{0.8cm}} + \underline{\hspace{0.8cm}} + \underline{\hspace{0.8cm}} = \underline{\hspace{1.5cm}}$ units

    Why is this not the total from item 41?

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    The shared edge is ______ units long, and it was counted ______ times instead of ______.

PAGE 13 — Tracing boundaries

Trace It, Don't Add It

43. Figure 1. $P = 10 + 3 + 4 + 3 + 6 + 6 = \underline{\hspace{1.5cm}}$ cm

FIGURE: fig6-semicircle-on-rectangle.png (half width)

44. Figure 6. Arc length of a semicircle $= \underline{\hspace{1.5cm}}$

    $P = 10 + 6 + 6 + \pi(\underline{\hspace{0.8cm}}) = \underline{\hspace{2cm}} \approx \underline{\hspace{2cm}}$ cm

    Which segment did you leave out, and why? _______________________________________________

FIGURE: fig9-semicircle-bite.png (half width)

45. Figure 9. $P = \underline{\hspace{2.5cm}} \approx \underline{\hspace{2.5cm}}$ cm

FIGURE: fig12-patio.png (half width)

46. Figure 12. $P = \underline{\hspace{2cm}}$ ft

PAGE 14 — Practice: find the perimeter

Perimeter Practice

FIGURE: fig15-practice-perimeter-set.png (full width)

47. Figure 15a. $P =$ _______________

48. Figure 15b. By tracing: $P =$ _______________

    The two rectangles apart: ______ $+$ ______ $=$ ______

    Difference: ______ Shared edge: ______ Is the difference twice the shared edge? ______

49. Figure 15c. $P =$ ____________ exactly, ____________ approximately

50. Figure 3. $P =$ _______________

51. Figure 4. $P =$ _______________

52. Figure 5. $P =$ _______________

PAGE 15 — Errors and reasoning

Interior Edges Do Not Count

53. Find the error. For Figure 15b a student answered $2(10 + 3) + 2(4 + 5) = 44$ cm.

    What went wrong? _______________________________________________

    Correct perimeter: _______________

54. Find the error. For Figure 15c a student answered $8 + 5 + 5 + 8 + 4\pi$.

    What went wrong? _______________________________________________

    Correct perimeter: _______________

55. Explain. Why does the sum of two pieces' perimeters always exceed the joined figure's perimeter by exactly twice the shared edge?

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56. Explain. Figure 1 has area $48$ cm$^2$ and perimeter $32$ cm. A $6$ cm by $8$ cm rectangle also has area $48$ cm$^2$.

    Its perimeter: ______ cm. What does this show?

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PAGE 16 — Exit ticket 12.3

Exit Ticket · Lesson 12.3

Name: ________________________ Date: ____________

57. Figure 15d. $P =$ _______________

58. Trim all the way around the window in Figure 11: ____________ exactly, ____________ ft to the nearest hundredth

59. Distance around the track in Figure 10: ____________ exactly, ____________ m approximately

60. Why is an interior subdivision line never part of a perimeter?

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PAGE 17 — Which measure does the problem want?

12.4 Composite Figures in Context

Complete the table.

Dollars per square foot multiplies an ____________. Dollars per foot multiplies a ____________.

FIGURE: fig12-patio.png (full width)

61. Paving at $9 per square foot.

    Area: ____________ ft$^2$ Cost: $____________

62. Trim for the window in Figure 11 at $3 per foot, sold in whole feet.

    Perimeter: ____________ ft Feet to buy: ______ Cost: $____________

PAGE 18 — The track and the field

Around It, and Across It

FIGURE: fig10-running-track.png (full width)

63. Distance around: $80 + 80 + 2\pi(\underline{\hspace{0.8cm}}) = \underline{\hspace{2cm}} \approx \underline{\hspace{2cm}}$ m

    Two laps: ____________ m

64. Area enclosed: $(80)(50) + \pi(\underline{\hspace{0.8cm}})^2 = \underline{\hspace{2cm}} \approx \underline{\hspace{2cm}}$ m$^2$

65. Figure 7. Seeded area: ____________ exactly, ____________ m$^2$ approximately

PAGE 19 — Practice in context

Real Jobs, Real Numbers

FIGURE: fig16-practice-context-set.png (full width)

66. Figure 16a. Area of the walk only: ____________ ft$^2$ At $8 per ft$^2$: $____________

67. Figure 16a. Railing around the outside edge: ____________ ft At $6 per ft: $____________

68. Figure 16b. Soil area: ____________ exactly, ____________ ft$^2$ approximately

    Edging: ____________ exactly, ____________ ft approximately

69. Figure 16c. Glass: ____________ exactly, ____________ ft$^2$ approximately

    Edge trim: ____________ exactly, ____________ ft approximately

PAGE 20 — Errors, reasoning, and a budget

Reading the Situation

70. Find the error. For Figure 16a a student found the walk's area as $(26)(18) = 468$ ft$^2$.

    What went wrong? _______________________________________________

    Correct area: ____________ ft$^2$

71. Find the error. For Figure 16b a student included the $8$ ft side in the edging.

    What went wrong? _______________________________________________

    Correct length: _______________

72. Explain. For Figure 16c, which measure buys the glass and which buys the trim? Give the units.

    Glass: ____________ , units ______ Trim: ____________ , units ______

73. Budget it. Garden bed in Figure 16b: soil $4 per ft$^2$, edging $3 per ft.

    Soil cost: $____________ Edging cost: $____________ Total: $____________

74. Explain. How do the units of a rate tell you whether the problem wants perimeter or area?

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PAGE 21 — Exit ticket 12.4

Exit Ticket · Lesson 12.4

Name: ________________________ Date: ____________

75. Sealer for the patio in Figure 12 at $0.75 per square foot: $____________

76. Distance in $2$ laps of the track in Figure 10: ____________ m

77. Edge trim for the tabletop in Figure 16c: ____________ exactly, ____________ ft approximately

78. Why is fencing measured in feet but sod in square feet?

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PAGE 22 — Review Part A

Chapter 12 Review · Area (8.MG.5a)

FIGURE: fig17-review-set.png (full width)

All measurements are in feet.

79. Figure 17a. Way 1: ____________ Way 2: ____________ Agree? ______

80. Figure 17b. Triangle base ______ height ______ Total area: ____________

81. Figure 17c. Exact ____________ Approximate ____________

82. Figure 17d. Trapezoid formula: ____________ Rectangle plus triangle: ____________

83. Figure 1 by subtraction: ____________

84. Figure 6: exact ____________ approximate ____________

85. Figure 7: exact ____________ approximate ____________

86. Figure 9: exact ____________ approximate ____________

87. Explain. Why must two different subdivisions of the same figure give the same area?

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88. Find the error. For Figure 17b a student used $13$ ft as the triangle's height.

    What went wrong? ____________________ Correct area: ____________

PAGE 23 — Review Part B

Chapter 12 Review · Perimeter (8.MG.5b)

Use the same review figures.

89. Figure 17a. $P =$ ____________

90. Figure 17b. $P =$ ____________

91. Figure 17c. Exact ____________ Approximate ____________

92. Figure 17d. $P =$ ____________

93. Figure 8. By tracing: ____________ By pieces, corrected: ______ $+$ ______ $- 2($ ______ $) =$ ____________

94. Figure 11. Exact ____________ Approximate ____________

95. Find the error. For Figure 17a a student added $2(18) + 2(13) = 62$ ft.

    What went wrong? _______________________________________________

    Correct perimeter: ____________

96. Explain. Why is a semicircle's arc part of the perimeter but its diameter is not?

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PAGE 24 — Review Part C

Chapter 12 Review · In Context (8.MG.5c)

97. Patio shaped like Figure 17a, paving $7 per ft$^2$. Area: ______ ft$^2$ Cost: $______

98. Edging for that patio at $2.50 per ft. Perimeter: ______ ft Cost: $______

99. Tent end shaped like Figure 17b. One pint of waterproofing covers $25$ ft$^2$.

    Area: ______ ft$^2$ Whole pints needed: ______

100. Countertop shaped like Figure 17c, sealer $1.20 per ft$^2$.

     Area: ______ ft$^2$ Cost: $______

101. Sign shaped like Figure 17d. Area: ______ ft$^2$ Trim: ______ ft

102. Track in Figure 10: resurfacing inside the boundary at $8 per m$^2$, and a rope once around at $1.50 per m.

     Resurfacing: $____________ Rope: $____________

103. Explain. A store sells fencing by the foot and sod by the square foot. Which measurement do you take for each, and why are the two answers different kinds of numbers?

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104. Find the error. A student priced window trim (Figure 11) using the area $20 + 2\pi$ ft$^2$ times $3 per foot.

     Two errors: _______________________________________________

     Correct cost, buying whole feet: $____________