Grade 8 Workbook — Chapter 11: Surface Area and Volume: Pyramids and Cones

SOL 8.MG.2 · Companion to Textbook Chapter 11

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 150.

Pi: use $\pi \approx 3.14$ everywhere in this chapter, not a calculator's stored $\pi$. Give every answer containing $\pi$ exactly first, then approximately.

Units: volume in cubic units (cm$^3$, in$^3$, ft$^3$, m$^3$); surface area in square units (cm$^2$, in$^2$, ft$^2$, m$^2$).

Formula reminders: $\ell^2 = h^2 + \left(\frac{s}{2}\right)^2$ · $SA = s^2 + 2s\ell$ · $LA = 2s\ell$ · $V = \frac{1}{3}s^2h$ · $V = \frac{1}{3}\pi r^2 h$

PAGE 1 — Chapter opener

Chapter 11 · Surface Area and Volume: Pyramids and Cones

Standard 8.MG.2

In this chapter you will:

• Name the parts of a square-based pyramid and a cone, and tell the height apart from the slant height
• Unfold a pyramid into a net and build the surface area formula from it
• Find a slant height from a height, and a height from a slant height
• Find the volume of square-based pyramids and cones
• Explain why a cone is one third of its cylinder and a pyramid is one third of its prism
• Decide whether a problem in context wants volume or surface area, and solve it

Words to know: square-based pyramid · base · apex · lateral face · base edge · height · slant height · net · surface area · lateral area · cone · radius · diameter · volume · cubic units · square units · base area $B$

Every pyramid in this chapter has a square base. Surface area is found for pyramids only — for cones this chapter finds volume and nothing else.

PAGE 2 — The parts of a square-based pyramid

11.1 Pyramids, Cones, and Two Different Heights

FIGURE: fig1-pyramid-parts.png (full width)

Fill in the blanks.

The point where all four triangles meet is the ____________.

The four triangular faces are called ____________ faces.

The height $h$ runs from the apex to the ____________ of the base.

The slant height $\ell$ runs from the apex to the ____________ of a base edge.

The longer of the two is always the ____________ ____________.

Volume uses the ____________. Surface area uses the ____________ ____________.

1. Name the shape of the base and the shape of each lateral face of a square-based pyramid.

   Base: _______________ Each lateral face: _______________

2. A square-based pyramid has ______ faces, ______ edges, and ______ vertices.

3. Which labeled segment in the figure above runs from the apex to the center of the base? ______

   Which runs from the apex to the midpoint of a base edge? ______

PAGE 3 — The net, and the cone

Nets and Cones

FIGURE: fig2-pyramid-net.png (full width)

4. In the net of a square-based pyramid there are ______ square(s) and ______ triangle(s).

FIGURE: fig3-cone-parts.png (full width)

13. A cone has base diameter $14$ cm and height $9$ cm.

    Radius: ______ cm Which given number is the height? ______ cm

14. Describe the net of a square-based pyramid, and say what the height of each triangle in the net represents.

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PAGE 4 — Finding the slant height

The Pythagorean Theorem Inside a Pyramid

FIGURE: fig4-slant-height-right-triangle.png (full width)

$$\ell^2 = h^2 + \left(\frac{s}{2}\right)^2 \qquad\qquad h^2 = \ell^2 - \left(\frac{s}{2}\right)^2$$

Warning: use half the base edge, never the whole base edge.

5. Base edge $6$ cm, height $4$ cm.

   $\ell^2 = \underline{\hspace{1.5cm}}^2 + \underline{\hspace{1.5cm}}^2 = \underline{\hspace{1.5cm}}$, so $\ell = \underline{\hspace{1.5cm}}$ cm.

6. Base edge $10$ in, height $12$ in. Slant height: ______ in

   WORK SPACE: 1.2 in tall, full width

7. Why can the slant height never equal the height?

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PAGE 5 — Independent practice 11.1

Practice · Heights and Slant Heights

8. Find the slant height of each square-based pyramid.

9. Base edge $18$ in, height $12$ in. Slant height: ______ in

10. Base edge $16$ cm, slant height $17$ cm. Height: ______ cm

11. Base edge $12$ m, slant height $10$ m. Height: ______ m

12. Base edge $10$ ft, slant height $13$ ft. Height: ______ ft

WORK SPACE: 2.0 in tall, full width

15. Find the error. For a pyramid with base edge $6$ cm and height $4$ cm, a student wrote $\ell^2 = 4^2 + 6^2 = 52$, so $\ell = \sqrt{52} \approx 7.2$ cm.

    What went wrong? _______________________________________________

    Correct slant height: ______ cm

16. Apply it. A glass paperweight is a square-based pyramid with base edge $16$ mm and height $15$ mm.

    Slant height: ______ mm

17. Explain. Why are all four lateral faces congruent isosceles triangles?

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PAGE 6 — Exit ticket 11.1

Exit Ticket · Lesson 11.1

Name: ________________________ Date: ____________

18. Base edge $14$ cm, height $24$ cm. Slant height: ______ cm

19. Base edge $30$ in, slant height $25$ in. Height: ______ in

20. In a net of a square-based pyramid, the height of each triangular face is called the ____________ ____________.

21. Explain the difference between the height and the slant height, and say which is longer.

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PAGE 7 — Building the surface area formula

11.2 Surface Area of a Square-Based Pyramid

FIGURE: fig5-surface-area-pieces.png (full width)

Build it from the net.

The base is a square with side $s$, so its area is ____________.

Each lateral face is a triangle with base $s$ and height ____________, so its area is ____________.

There are ______ lateral faces.

22. $SA = s^2 + 4\left(\frac{1}{2}s\ell\right) = s^2 + \underline{\hspace{2cm}}$

$$SA = s^2 + 2s\ell \qquad\qquad LA = 2s\ell$$

Lateral area only is what you want when the base is not covered: a tent with no floor, shingles on a roof, paint on the four sloped sides.

PAGE 8 — Guided practice 11.2

Practice · Surface Area

27. Lateral area only: base edge $6$ cm, slant height $8$ cm. $LA =$ ______ cm$^2$

28. Base edge $6$ m and height $4$ m.

    Step 1 — slant height: ______ m Step 2 — surface area: ______ m$^2$

    WORK SPACE: 1.4 in tall, full width

29. Base edge $12$ in, slant height $10$ in. $SA =$ ______ in$^2$

PAGE 9 — Independent practice 11.2

Practice · Surface Area and Lateral Area

30. Find each surface area.

31. Base edge $14$ ft, slant height $25$ ft. $SA =$ ______ ft$^2$

32. Base edge $20$ cm, slant height $15$ cm. $SA =$ ______ cm$^2$

33. Lateral area: base edge $18$ m, slant height $15$ m. $LA =$ ______ m$^2$

These three give the height. Find the slant height first.

34. Base edge $16$ in, height $15$ in. $\ell =$ ______ in $SA =$ ______ in$^2$

35. Base edge $12$ cm, height $8$ cm. $\ell =$ ______ cm $SA =$ ______ cm$^2$

36. Base edge $24$ ft, height $5$ ft. $\ell =$ ______ ft $SA =$ ______ ft$^2$

WORK SPACE: 2.2 in tall, full width

37. Base edge $7$ m, surface area $189$ m$^2$. Slant height: ______ m

PAGE 10 — Applications and errors 11.2

Apply and Check

38. Apply it. A tent is a square-based pyramid with base edge $6$ ft and slant height $5$ ft. Four fabric walls, no floor.

    Volume or surface area? ____________ Is the base included? ______

    Fabric needed: ______ ft$^2$

39. Apply it. A gift box is a square-based pyramid with base edge $10$ cm and height $12$ cm. Paper covers the entire outside.

    Slant height: ______ cm Paper needed: ______ cm$^2$

    WORK SPACE: 1.4 in tall, full width

40. Find the error. For a pyramid with base edge $6$ cm and height $4$ cm, a student wrote $SA = 6^2 + 2(6)(4) = 84$ cm$^2$.

    What went wrong? _______________________________________________

    Why must the student's answer be too small? _______________________________________________

    Correct surface area: ______ cm$^2$

41. Explain. Why does the formula contain $2s\ell$ and not $4s\ell$, when there are four lateral faces?

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PAGE 11 — Exit ticket 11.2

Exit Ticket · Lesson 11.2

Name: ________________________ Date: ____________

42. Base edge $5$ cm, slant height $6$ cm. $SA =$ ______ cm$^2$

43. Base edge $2$ m, slant height $5$ m. $SA =$ ______ m$^2$

44. Base edge $16$ in, height $6$ in. $SA =$ ______ in$^2$

45. Lateral area: base edge $12$ ft, slant height $13$ ft. $LA =$ ______ ft$^2$

46. Why can the pyramid's height not be substituted into the surface area formula, and what do you do when a problem gives you the height?

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PAGE 12 — The one-third relationship

11.3 Why One Third?

FIGURE: fig6-cone-in-cylinder.png (half width, left)

FIGURE: fig7-pyramid-in-prism.png (half width, right)

FIGURE: fig8-three-pours.png (full width)

Fill in the blanks.

A cylinder holds $V = \pi r^2 h$. A cone with the same base and the same height holds ____________ of that.

A prism with a square base holds $V = s^2h$. A square-based pyramid with the same base and the same height holds ____________ of that.

One formula covers both: $V = \underline{\hspace{2cm}}Bh$, where $B$ is the ____________ of the base.

It takes ______ cone-fulls to fill the cylinder.

47. A cone holds one third of a cylinder when the two solids have the same ____________ and the same ____________.

48. How many cone-fulls fill a cylinder with the same base and the same height? ______

49. In the pouring figure, what fraction is filled after two pours? ______

PAGE 13 — Guided practice 11.3

Practice · Reasoning Up and Down

53. Explain what "the same base and the same height" means for a cone and a cylinder. Name the two measurements that must match.

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PAGE 14 — Independent practice 11.3

Practice · Cones with Cylinders, Pyramids with Prisms

55. Cylinder: radius $6$ cm, height $10$ cm. $V =$ ______ Cone with the same base and height: ______ , or about ______

56. Cylinder: radius $2$ in, height $6$ in. $V =$ ______ Cone: ______ , or about ______

57. Prism: square base side $9$ m, height $4$ m. $V =$ ______ m$^3$ Pyramid: ______ m$^3$

58. Prism: square base side $12$ ft, height $5$ ft. $V =$ ______ ft$^3$ Pyramid: ______ ft$^3$

59. A cone holds $30\pi$ cm$^3$. The cylinder with the same base and height holds ______ , or about ______

60. A pyramid holds $50$ in$^3$. The prism with the same base and height holds ______ in$^3$

WORK SPACE: 2.0 in tall, full width

PAGE 15 — Reasoning 11.3

Explain It

61. Explain. A cube with edge $6$ cm cuts into three congruent square-based pyramids, each with a face of the cube as its base and height $6$ cm.

    Volume of the cube: ______ cm$^3$ Volume of one pyramid: ______ cm$^3$

    Check with $\frac{1}{3}s^2h$: _______________

62. Explain. Why does a cone hold less than half of its cylinder? Use horizontal cross-sections.

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63. Explain. A cone and a cylinder have the same height, but the cone's radius is twice the cylinder's radius $R$. Is the cone one third of the cylinder?

    Cone volume in terms of $R$ and $h$: _______________ Cylinder volume: _______________

    Answer and why: _______________________________________________

64. Apply it. A cylindrical container holds $900$ mL. A cone with the same base and height holds ______ mL

65. Find the error. A student says a pyramid is half of its prism "because a triangle is half of a rectangle."

    Why does the two-dimensional fact not carry over? _______________________________________________

    Correct fraction: ______

66. Explain. Why does $V = \frac{1}{3}Bh$ work for both a cone and a pyramid? What is $B$ for each?

    Cone: $B =$ ____________ Pyramid: $B =$ ____________

PAGE 16 — Exit ticket 11.3

Exit Ticket · Lesson 11.3

Name: ________________________ Date: ____________

67. A cylinder holds $48\pi$ m$^3$. The cone with the same base and height holds ______ , or about ______

68. A pyramid holds $21$ cm$^3$. The prism with the same base and height holds ______ cm$^3$

69. How many pyramid-fulls fill a prism with the same square base and height? ______

70. Explain why a cone holds exactly one third of its cylinder. Use both "same base" and "same height."

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PAGE 17 — The volume formulas

11.4 Volume of Square-Based Pyramids and Cones

$$V = \frac{1}{3}s^2h \qquad\qquad V = \frac{1}{3}\pi r^2 h$$

The $h$ is the height — never the slant height.

Checklist before you compute. Square first, then multiply. Keep the $\frac{1}{3}$. Halve a diameter. Write cubic units. Approximate $\pi$ only at the last step.

71. Base edge $3$ cm, height $5$ cm: $V = \frac{1}{3}(\underline{\hspace{1.5cm}})^2(\underline{\hspace{1.5cm}}) = \underline{\hspace{1.5cm}}$ cm$^3$

PAGE 18 — Guided practice 11.4

Practice · Volume

Pyramids.

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

WORK SPACE: 2.0 in tall, full width

PAGE 19 — Independent practice 11.4

Practice · Volume, Mixed

79. Find each pyramid volume.

80. Pyramid: base edge $7$ ft, height $6$ ft. $V =$ ______ ft$^3$

81. Pyramid: base edge $20$ cm, height $6$ cm. $V =$ ______ cm$^3$

82. Cone: radius $6$ cm, height $4$ cm. $V =$ ______ , or about ______

83. Cone: diameter $8$ in, height $6$ in. $r =$ ______ in $V =$ ______ , or about ______

84. Cone: radius $9$ m, height $5$ m. $V =$ ______ , or about ______

85. Cone: radius $1$ ft, height $9$ ft. $V =$ ______ , or about ______

WORK SPACE: 2.2 in tall, full width

PAGE 20 — Working backward, and applications 11.4

Work Backward and Apply

86. Pyramid: base edge $12$ ft, volume $240$ ft$^3$. Height: ______ ft

    WORK SPACE: 1.0 in tall, full width

87. Cone: radius $5$ in, volume $100\pi$ in$^3$. Height: ______ in

88. Pyramid: base edge $6$ m, slant height $5$ m.

    Step 1 — height: ______ m Step 2 — volume: ______ m$^3$

89. Cone: base area $27\pi$ cm$^2$, height $5$ cm. Using $V = \frac{1}{3}Bh$: ______ , or about ______

90. Apply it. A cone-shaped paper cup has radius $3$ cm and height $10$ cm. It holds ______ , or about ______

91. Apply it. A candle mold is a square-based pyramid with base edge $10$ cm and height $12$ cm. It holds ______ cm$^3$

92. Find the error. For a cone with radius $6$ cm and height $10$ cm, a student wrote $V = \pi(6)^2(10) = 360\pi$ cm$^3$.

    What went wrong? _______________________________________________

    Correct volume: ______ , or about ______

PAGE 21 — Exit ticket 11.4

Exit Ticket · Lesson 11.4

Name: ________________________ Date: ____________

93. Pyramid: base edge $11$ cm, height $3$ cm. $V =$ ______ cm$^3$

94. Pyramid: base edge $2$ m, height $9$ m. $V =$ ______ m$^3$

95. Cone: radius $6$ in, height $9$ in. $V =$ ______ , or about ______

96. Cone: diameter $6$ ft, height $8$ ft. $V =$ ______ , or about ______

97. Where does the $\frac{1}{3}$ in both formulas come from, and why is it the height and not the slant height in them?

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PAGE 22 — Deciding what a problem wants

11.5 Problems in Context

Volume is about the inside: fill, hold, pour, capacity, how much sand, water, wax, grain. → cubic units

Surface area is about the outside: cover, wrap, paint, tile, how much fabric, paper, glass. → square units

Then ask: is the base included?

• closed box, glass case, painted model → all five faces, $SA = s^2 + 2s\ell$
• tent with no floor, roof shingles, four sloped sides → four faces, $LA = 2s\ell$

Four moves for every context problem. 1. Decide. 2. Collect (halve a diameter; find $\ell$ or $h$ if needed). 3. Substitute. 4. Report with the right unit.

98. Volume or surface area?

PAGE 23 — Guided practice 11.5

Practice · In Context

99. A sandbox is a square-based pyramid with base edge $12$ cm and height $8$ cm.

    Volume or surface area? ____________ Answer: ______ cm$^3$

100. An ornament is a square-based pyramid with base edge $10$ cm and height $12$ cm. Foil covers all five faces.

     Slant height: ______ cm Answer: ______ cm$^2$

101. A snow cone cup has radius $3$ cm and height $8$ cm. It holds ______ , or about ______

102. A paperweight is a square-based pyramid with base edge $4$ cm and slant height $6$ cm. A label covers the four lateral faces only.

     Answer: ______ cm$^2$

103. A cone-shaped cup has radius $2$ in and height $9$ in. It holds ______ , or about ______

WORK SPACE: 2.0 in tall, full width

PAGE 24 — Independent practice 11.5, part 1

Apply It

104. Apply it. A roof is a square-based pyramid with base edge $18$ ft and slant height $15$ ft. Shingles cover the four sloped faces only. ______ ft$^2$

105. Apply it. A gravel pile is a cone with radius $10$ ft and height $6$ ft. ______ , or about ______

106. Apply it. A display case is a square-based pyramid with base edge $16$ in and height $15$ in, glass on all five faces.

     Slant height: ______ in Glass: ______ in$^2$

107. Apply it. A cone-shaped filter has base diameter $12$ cm and height $7$ cm. $r =$ ______ cm It holds ______ , or about ______

108. Apply it. A candle is a square-based pyramid with base edge $6$ cm and height $4$ cm. Wax has mass $0.9$ g per cm$^3$.

     Volume: ______ cm$^3$ Mass: ______ g

WORK SPACE: 2.2 in tall, full width

PAGE 25 — Independent practice 11.5, part 2

Apply It (continued)

109. Apply it. Which holds more?

 Which holds more, and by about how much? _______________________________________________

110. Apply it. A cone-shaped funnel has radius $5$ cm and height $12$ cm and drains at $20$ cm$^3$ per second.

     Volume: ______ , or about ______ Time to empty: about ______ seconds

111. Apply it. A tent is a square-based pyramid with base edge $8$ ft and height $3$ ft, four walls and no floor.

     Slant height: ______ ft Fabric: ______ ft$^2$

112. Apply it. A cone-shaped hole has radius $3$ ft and depth $4$ ft. Mulch costs $3 per cubic foot.

     Volume: ______ , or about ______ ft$^3$ Cost: $______

113. Explain. A problem gives a pyramid's base edge and slant height and asks for volume. What must you find first, and why?

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114. Find the error. For wrapping paper on a closed pyramid box with base edge $10$ in and height $12$ in, a student wrote $SA = 10^2 + 2(10)(12) = 340$ in$^2$.

     What went wrong? _______________________________________________

     Correct surface area: ______ in$^2$

115. Explain. How does the unit on your answer tell you which kind of problem you solved?

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PAGE 26 — Exit ticket 11.5

Exit Ticket · Lesson 11.5

Name: ________________________ Date: ____________

116. A cone-shaped cup has radius $4$ cm and height $9$ cm. It holds ______ , or about ______

117. A square-based pyramid has base edge $24$ in and height $5$ in. Sand fills it: ______ in$^3$

118. A square-based pyramid has base edge $20$ cm and slant height $26$ cm. Paper covering all five faces: ______ cm$^2$

119. Cardboard for a closed pyramid box with base edge $6$ cm and slant height $8$ cm: ______ cm$^2$

120. How do you decide whether a context problem wants volume or surface area?

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PAGE 27 — Chapter 11 review, Part A

Chapter 11 Review

Part A · Surface area of square-based pyramids (8.MG.2a)

121. Describe the net of a square-based pyramid: how many pieces of each shape, and what the height of each triangle represents.

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122. Base edge $48$ cm, height $7$ cm. Slant height: ______ cm

123. Find each surface area.

124. Base edge $30$ ft, height $20$ ft. $\ell =$ ______ ft $SA =$ ______ ft$^2$

125. Lateral area: base edge $14$ in, slant height $25$ in. $LA =$ ______ in$^2$

126. Base edge $24$ in, surface area $1200$ in$^2$. Slant height: ______ in

127. Find the error. For base edge $6$ cm and slant height $5$ cm, a student wrote $36 + 3(15) = 81$ cm$^2$.

     What went wrong? _______________________________________________ Correct: ______ cm$^2$

128. Explain. How does the net show that $SA = s^2 + 2s\ell$? Account for every piece and for the $2$.

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PAGE 28 — Chapter 11 review, Part B

Chapter 11 Review (continued)

Part B · Volume of cones and square-based pyramids (8.MG.2b)

Give every cone volume exactly, then approximately with $\pi \approx 3.14$.

129. Find each pyramid volume.

130. Pyramid: base edge $16$ ft, height $15$ ft. $V =$ ______ ft$^3$

131. Find each cone volume.

132. Cone: diameter $10$ m, height $12$ m. $V =$ ______ , or about ______

133. Cone: radius $12$ ft, height $5$ ft. $V =$ ______ , or about ______

134. Pyramid: base edge $15$ m, volume $600$ m$^3$. Height: ______ m

135. Cone: height $4$ cm, volume $48\pi$ cm$^3$. Radius: ______ cm

136. Pyramid: base edge $12$ cm, slant height $10$ cm. $h =$ ______ cm $V =$ ______ cm$^3$

WORK SPACE: 2.2 in tall, full width

PAGE 29 — Chapter 11 review, Part C

Chapter 11 Review (continued)

Part C · Explaining the volume relationships (8.MG.2c)

137. Cylinder: radius $5$ cm, height $9$ cm. $V =$ ______ Cone with the same base and height: ______ , or about ______

138. Prism: square base side $10$ in, height $9$ in. $V =$ ______ in$^3$ Pyramid: ______ in$^3$

139. A cone holds $16\pi$ ft$^3$. The cylinder with the same base and height holds ______ ft$^3$

140. Explain. Use the dissection of a cube into three congruent square-based pyramids to explain the one-third relationship.

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141. Explain. Describe the pouring experiment and say exactly what it shows. Why are both "same base" and "same height" required?

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142. Explain. A classmate writes, "A cone is one third of any cylinder." Correct the statement, then give a counterexample with numbers.

     Correction: _______________________________________________

     Counterexample: cone $r =$ ______ , $h =$ ______ , $V =$ ______ ; cylinder $r =$ ______ , $h =$ ______ , $V =$ ______

PAGE 30 — Chapter 11 review, Part D

Chapter 11 Review (continued)

Part D · Problems in context (8.MG.2d)

Give every cone volume exactly, then approximately with $\pi \approx 3.14$.

143. Apply it. A cone-shaped cup has radius $4$ cm and height $6$ cm. It holds ______ , or about ______

144. Apply it. A gift box is a square-based pyramid with base edge $18$ cm and height $12$ cm. Paper for the whole outside: $\ell =$ ______ cm, $SA =$ ______ cm$^2$

145. Apply it. A sand pile is a cone with base diameter $14$ ft and height $3$ ft. $r =$ ______ ft $V =$ ______ , or about ______

146. Apply it. A monument is a square-based pyramid with base edge $30$ m and slant height $25$ m. Paint covers the four sloped faces only: ______ m$^2$

147. Apply it. A cone-shaped funnel has radius $6$ cm and height $10$ cm.

     Volume: ______ , or about ______ Full $100$ cm$^3$ jars it can fill: ______

148. Apply it. A planter is a square-based pyramid with base edge $12$ in and height $8$ in.

     a) Soil it holds: ______ in$^3$

     b) Liner for the four lateral faces only: $\ell =$ ______ in, $LA =$ ______ in$^2$

WORK SPACE: 2.0 in tall, full width

149. Explain. A square-based pyramid has base edge $6$ cm and height $4$ cm.

     Volume: ______ cm$^3$ Surface area: ______ cm$^2$

     Why do the two answers carry different units? _______________________________________________

150. Write your own. Write a context problem about a cone that must be solved with volume, and one about a square-based pyramid that must be solved with surface area. Then solve both.

     WORK SPACE: 3.5 in tall, full width