Kenneth ChengMathematical modeling
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Surface areas and volumes of simple solids

Read the idea, work independently, then explain what changed.

高一必修 第二册(A版).pdf · 8.3 · PDF 121 / printed page 114

Revisit first: Oblique drawings of solids

TOPIC 01

Surface areas and volumes of simple solids

Build understanding of surface areas and volumes of simple solids through definitions, contrasting cases and justified applications.

What you will be able to explain

  • Use definitions, valid spatial reasoning and metric checks for surface areas and volumes of simple solids.
  • Connect representations and justify the steps, including boundary cases.
  • Explain a solution and apply the idea to a changed situation.

Area and volume

Count actual boundary faces for surface area and use perpendicular heights for volume.

Similarity scales

Length, area and volume use first, second and third powers of the linear scale.

PREDICT → EXPLORE → EXPLAIN → TRANSFER

Make a prediction, then explore the relationship.

Lesson question: Before calculating, predict how the conclusion changes when one defining condition in surface areas and volumes of simple solids changes. Record a reason.

Model exploration: predict which displayed result changes with the parameters, check the values, and compare the observation with the lesson question.

Schematic cuboid (not to scale): 3×4×2; volume=24, surface area=52.

Schematic cuboid (not to scale): 3×4×2; volume=24, surface area=52.

Explain: Compare two admissible cases and one boundary or invalid case. Explain the observed difference using the stated definition.

Transfer: Construct a new example and a tempting incorrect solution. Repair the solution by naming the missing condition.

Try it. Leave your reasoning visible.

Use one hint at a time. A correction explains what changed, not just the final answer.

01 / Foundation#Worked example

Find the volume of the prism.

Sbase=4,h=3S_{\text{base}}=4,\quad h=3
  • Use the domain, units and sampling assumptions stated in the question.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Choose the correct surface or volume formula and keep units and similarity powers consistent.
Hint 2
Use perpendicular height, not a slant edge.
Worked solution

  1. Choose the correct surface or volume formula and keep units and similarity powers consistent.

  2. Calculate or simplify this relation.

    V=Sh=4⋅3=12V=Sh=4\cdot3=12

  3. Base area times perpendicular height gives prism volume.

The requested value is 12.

Checks and common pitfalls: Base area times perpendicular height gives prism volume.

Think first. Reveal a hint when the class is ready.

02 / Standard#Worked example

A cone has radius r and height h. Write its volume as cπ and find c.

r=2,h=3r=2,\quad h=3
  • Use the domain, units and sampling assumptions stated in the question.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Choose the correct surface or volume formula and keep units and similarity powers consistent.
Hint 2
A cone occupies one third of the corresponding cylinder volume.
Worked solution

  1. Choose the correct surface or volume formula and keep units and similarity powers consistent.

  2. Calculate or simplify this relation.

    V=13πr2h=4πV=\frac13\pi r^2h=4\pi

  3. Do not omit the factor one third.

The requested value is 4.

Checks and common pitfalls: Do not omit the factor one third.

Think first. Reveal a hint when the class is ready.

03 / Transfer#Worked example

Similar solids have volume ratio 64 from large to small. Find the linear scale factor.

  • Use the domain, units and sampling assumptions stated in the question.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Choose the correct surface or volume formula and keep units and similarity powers consistent.
Hint 2
Take the positive cube root.
Worked solution

  1. Choose the correct surface or volume formula and keep units and similarity powers consistent.

  2. Calculate or simplify this relation.

    λ3=64⇒λ=4\lambda^3=64\Rightarrow\lambda=4

  3. Lengths, areas and volumes scale with powers one, two and three respectively.

The requested value is 4.

Checks and common pitfalls: Lengths, areas and volumes scale with powers one, two and three respectively.

Think first. Reveal a hint when the class is ready.

04 / Foundation#Your turn

Find the volume of the prism.

Sbase=5,h=3S_{\text{base}}=5,\quad h=3
  • Use the domain, units and sampling assumptions stated in the question.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Choose the correct surface or volume formula and keep units and similarity powers consistent.
Hint 2
Use perpendicular height, not a slant edge.
Worked solution

  1. Choose the correct surface or volume formula and keep units and similarity powers consistent.

  2. Calculate or simplify this relation.

    V=Sh=5⋅3=15V=Sh=5\cdot3=15

  3. Base area times perpendicular height gives prism volume.

The requested value is 15.

Checks and common pitfalls: Base area times perpendicular height gives prism volume.

Think first. Reveal a hint when the class is ready.

05 / Foundation#Your turn

A cone has radius r and height h. Write its volume as cπ and find c.

r=3,h=3r=3,\quad h=3
  • Use the domain, units and sampling assumptions stated in the question.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Choose the correct surface or volume formula and keep units and similarity powers consistent.
Hint 2
A cone occupies one third of the corresponding cylinder volume.
Worked solution

  1. Choose the correct surface or volume formula and keep units and similarity powers consistent.

  2. Calculate or simplify this relation.

    V=13πr2h=9πV=\frac13\pi r^2h=9\pi

  3. Do not omit the factor one third.

The requested value is 9.

Checks and common pitfalls: Do not omit the factor one third.

Think first. Reveal a hint when the class is ready.

06 / Foundation#Your turn

A sphere has radius r. Write its volume as cπ and find c.

r=3r=3
  • Use the domain, units and sampling assumptions stated in the question.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Choose the correct surface or volume formula and keep units and similarity powers consistent.
Hint 2
Use the cube of the radius.
Worked solution

  1. Choose the correct surface or volume formula and keep units and similarity powers consistent.

  2. Calculate or simplify this relation.

    V=43πr3=1083πV=\frac43\pi r^3=\frac{108}3\pi

  3. Diameter must be halved before substitution if it is the given measurement.

The requested value is 36.

Checks and common pitfalls: Diameter must be halved before substitution if it is the given measurement.

Think first. Reveal a hint when the class is ready.

07 / Foundation#Your turn

Find c when the total surface area of this closed cylinder is cπ.

r=3,h=4r=3,\quad h=4
  • Use the domain, units and sampling assumptions stated in the question.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Choose the correct surface or volume formula and keep units and similarity powers consistent.
Hint 2
Include both circular ends and the curved side.
Worked solution

  1. Choose the correct surface or volume formula and keep units and similarity powers consistent.

  2. Calculate or simplify this relation.

    S=2πr2+2πrh=42πS=2\pi r^2+2\pi rh=42\pi

  3. An open cylinder would require a different boundary-area count.

The requested value is 42.

Checks and common pitfalls: An open cylinder would require a different boundary-area count.

Think first. Reveal a hint when the class is ready.

08 / Standard#Your turn

Find the volume of a pyramid frustum with parallel bases.

S1=9,S2=36,h=3S_1=9,\quad S_2=36,\quad h=3
  • Use the domain, units and sampling assumptions stated in the question.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Choose the correct surface or volume formula and keep units and similarity powers consistent.
Hint 2
The geometric-mean term links the similar bases.
Worked solution

  1. Choose the correct surface or volume formula and keep units and similarity powers consistent.

  2. Calculate or simplify this relation.

    V=h3(S1+S2+S1S2)=63V=\frac h3(S_1+S_2+\sqrt{S_1S_2})=63

  3. The formula assumes a frustum formed by parallel cutting of a pyramid.

The requested value is 63.

Checks and common pitfalls: The formula assumes a frustum formed by parallel cutting of a pyramid.

Think first. Reveal a hint when the class is ready.

09 / Standard#Your turn

Similar solids have linear scale factor 3 from small to large. Find the surface-area ratio.

  • Use the domain, units and sampling assumptions stated in the question.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Choose the correct surface or volume formula and keep units and similarity powers consistent.
Hint 2
Area scales in two dimensions.
Worked solution

  1. Choose the correct surface or volume formula and keep units and similarity powers consistent.

  2. Calculate or simplify this relation.

    Slarge/Ssmall=32=9S_{\text{large}}/S_{\text{small}}=3^2=9

  3. The volume ratio would instead be 27.

The requested value is 9.

Checks and common pitfalls: The volume ratio would instead be 27.

Think first. Reveal a hint when the class is ready.

10 / Standard#Your turn

A sphere is replaced by eight similar smaller spheres with the same total volume. Find the ratio of total new surface area to the original area.

  • Use the domain, units and sampling assumptions stated in the question.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Use volume conservation to find the linear scale first.
Hint 2
Each new radius is half the original radius.
Worked solution

  1. Use volume conservation to find the linear scale first.

  2. Calculate or simplify this relation.

    8r3=R3⇒r=R/2;8r2R2=28r^3=R^3\Rightarrow r=R/2;\quad\frac{8r^2}{R^2}=2

  3. Equal total volume does not imply equal total surface area.

The requested value is 2.

Checks and common pitfalls: Equal total volume does not imply equal total surface area.

Think first. Reveal a hint when the class is ready.

11 / Standard#Your turn

Find the volume of the prism.

Sbase=6,h=3S_{\text{base}}=6,\quad h=3
  • Use the domain, units and sampling assumptions stated in the question.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Choose the correct surface or volume formula and keep units and similarity powers consistent.
Hint 2
Use perpendicular height, not a slant edge.
Worked solution

  1. Choose the correct surface or volume formula and keep units and similarity powers consistent.

  2. Calculate or simplify this relation.

    V=Sh=6⋅3=18V=Sh=6\cdot3=18

  3. Base area times perpendicular height gives prism volume.

The requested value is 18.

Checks and common pitfalls: Base area times perpendicular height gives prism volume.

Think first. Reveal a hint when the class is ready.

12 / Transfer#Your turn

A cone has radius r and height h. Write its volume as cπ and find c.

r=4,h=3r=4,\quad h=3
  • Use the domain, units and sampling assumptions stated in the question.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Choose the correct surface or volume formula and keep units and similarity powers consistent.
Hint 2
A cone occupies one third of the corresponding cylinder volume.
Worked solution

  1. Choose the correct surface or volume formula and keep units and similarity powers consistent.

  2. Calculate or simplify this relation.

    V=13πr2h=16πV=\frac13\pi r^2h=16\pi

  3. Do not omit the factor one third.

The requested value is 16.

Checks and common pitfalls: Do not omit the factor one third.

Think first. Reveal a hint when the class is ready.

13 / Transfer#Your turn

A sphere has radius r. Write its volume as cπ and find c.

r=4r=4
  • Use the domain, units and sampling assumptions stated in the question.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Choose the correct surface or volume formula and keep units and similarity powers consistent.
Hint 2
Use the cube of the radius.
Worked solution

  1. Choose the correct surface or volume formula and keep units and similarity powers consistent.

  2. Calculate or simplify this relation.

    V=43πr3=2563πV=\frac43\pi r^3=\frac{256}3\pi

  3. Diameter must be halved before substitution if it is the given measurement.

The requested value is 85.33333333.

Checks and common pitfalls: Diameter must be halved before substitution if it is the given measurement.

Think first. Reveal a hint when the class is ready.

Focus on one question

End-of-lesson check and correction

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    Teacher preparation and assessment

    Question sequence

    • What must be true before using the main rule for surface areas and volumes of simple solids?
    • Which representation makes this task easier, and why?
    • Change one assumption. Does the conclusion survive?

    Board plan

    • Use definitions, valid spatial reasoning and metric checks for surface areas and volumes of simple solids.
    • Area and volume: Count actual boundary faces for surface area and use perpendicular heights for volume.
    • Similarity scales: Length, area and volume use first, second and third powers of the linear scale.
    • Close with: conditions → representation → reasoning → check.

    Anticipated thinking

    • Expected reasoning: Count actual boundary faces for surface area and use perpendicular heights for volume.
    • Expected reasoning: Length, area and volume use first, second and third powers of the linear scale.
    • Expected correction: A cone or pyramid needs the factor one third.

    Assessment checklist

    • 1: identify the givens and required quantity.
    • 1: choose a valid definition, representation or method.
    • 1: present connected, correct reasoning.
    • 1: check conditions and explain the result.

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    Curriculum and source notes ↗