Kenneth ChengMathematical modeling
← Senior Mathematics Studio

LEARN · EXPLAIN · REVISE

Basic solid figures

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

高一必修 第二册(A版).pdf · 8.1 · PDF 104 / printed page 97

Revisit first: Applications of plane vectors

TOPIC 01

Basic solid figures

Build understanding of basic solid figures through definitions, contrasting cases and justified applications.

What you will be able to explain

  • Use definitions, valid spatial reasoning and metric checks for basic solid figures.
  • Connect representations and justify the steps, including boundary cases.
  • Explain a solution and apply the idea to a changed situation.

Defining structures

Prisms, pyramids, frustums and solids of revolution have different defining base and side relations.

Sections

Parallel sections of a pyramid are similar; metric ratios depend on distance from the apex.

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 basic solid figures 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

How many vertices does an 5-gonal prism have?

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

Working and explanation

BUILD THE REASONING

Hint 1
Identify the defining faces, cross-sections and generating elements of the solid.
Hint 2
Each base contributes n vertices.
Worked solution

  1. Identify the defining faces, cross-sections and generating elements of the solid.

  2. Calculate or simplify this relation.

    V=2n=2(5)=10V=2n=2(5)=10

  3. Corresponding base vertices are distinct and joined by lateral edges.

The requested value is 10.

Checks and common pitfalls: Corresponding base vertices are distinct and joined by lateral edges.

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

02 / Standard#Worked example

How many edges does an 5-gonal pyramid have?

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

Working and explanation

BUILD THE REASONING

Hint 1
Identify the defining faces, cross-sections and generating elements of the solid.
Hint 2
Count base edges and apex-to-base edges separately.
Worked solution

  1. Identify the defining faces, cross-sections and generating elements of the solid.

  2. Calculate or simplify this relation.

    E=n+n=2(5)=10E=n+n=2(5)=10

  3. Do not count diagonals of the base as edges of the solid.

The requested value is 10.

Checks and common pitfalls: Do not count diagonals of the base as edges of the solid.

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

03 / Transfer#Worked example

Verify Euler’s relation for an 5-gonal prism by giving V,E,F.

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

Working and explanation

BUILD THE REASONING

Hint 1
Identify the defining faces, cross-sections and generating elements of the solid.
Hint 2
Count two bases and n lateral faces.
Worked solution

  1. Identify the defining faces, cross-sections and generating elements of the solid.

  2. Calculate or simplify this relation.

    2n−3n+(n+2)=2(n=5)2n-3n+(n+2)=2\quad(n=5)

  3. This verifies the relation for this convex polyhedron family; it is not a proof for every polyhedron.

V=10, E=15, F=7; V−E+F=2.

Checks and common pitfalls: This verifies the relation for this convex polyhedron family; it is not a proof for every polyhedron.

Reasoning checklist · self / teacher assessment
  • State the relevant definition, condition or model.
  • Show a valid calculation, proof or counterexample.
  • Interpret the conclusion with its restrictions.

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

04 / Foundation#Your turn

How many vertices does an 6-gonal prism have?

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

Working and explanation

BUILD THE REASONING

Hint 1
Identify the defining faces, cross-sections and generating elements of the solid.
Hint 2
Each base contributes n vertices.
Worked solution

  1. Identify the defining faces, cross-sections and generating elements of the solid.

  2. Calculate or simplify this relation.

    V=2n=2(6)=12V=2n=2(6)=12

  3. Corresponding base vertices are distinct and joined by lateral edges.

The requested value is 12.

Checks and common pitfalls: Corresponding base vertices are distinct and joined by lateral edges.

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

05 / Foundation#Your turn

How many edges does an 6-gonal pyramid have?

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

Working and explanation

BUILD THE REASONING

Hint 1
Identify the defining faces, cross-sections and generating elements of the solid.
Hint 2
Count base edges and apex-to-base edges separately.
Worked solution

  1. Identify the defining faces, cross-sections and generating elements of the solid.

  2. Calculate or simplify this relation.

    E=n+n=2(6)=12E=n+n=2(6)=12

  3. Do not count diagonals of the base as edges of the solid.

The requested value is 12.

Checks and common pitfalls: Do not count diagonals of the base as edges of the solid.

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

06 / Foundation#Your turn

Find the slant height of the right circular cone.

r=9,h=12r=9,\quad h=12
  • Use the domain, units and sampling assumptions stated in the question.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Identify the defining faces, cross-sections and generating elements of the solid.
Hint 2
The radius and vertical height form a right triangle.
Worked solution

  1. Identify the defining faces, cross-sections and generating elements of the solid.

  2. Calculate or simplify this relation.

    l=r2+h2=81+144=15l=\sqrt{r^2+h^2}=\sqrt{81+144}=15

  3. A slant height is not the perpendicular height.

The requested value is 15.

Checks and common pitfalls: A slant height is not the perpendicular height.

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

07 / Foundation#Your turn

A plane parallel to a pyramid base cuts halfway between apex and base. Find the ratio of the small cross-section area to the base area.

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

Working and explanation

BUILD THE REASONING

Hint 1
Identify the defining faces, cross-sections and generating elements of the solid.
Hint 2
Parallel sections form similar figures.
Worked solution

  1. Identify the defining faces, cross-sections and generating elements of the solid.

  2. Calculate or simplify this relation.

    λ=1/2;Ssection/Sbase=λ2=1/4\lambda=1/2;\quad S_{\text{section}}/S_{\text{base}}=\lambda^2=1/4

  3. Area ratios are squares of linear ratios.

The requested value is 0.25.

Checks and common pitfalls: Area ratios are squares of linear ratios.

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

08 / Standard#Your turn

Does having two parallel faces alone guarantee a prism? Explain.

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

Working and explanation

BUILD THE REASONING

Hint 1
Identify the defining faces, cross-sections and generating elements of the solid.
Hint 2
Consider a frustum cut parallel to a pyramid base.
Worked solution

  1. Identify the defining faces, cross-sections and generating elements of the solid.

  2. A truncated pyramid can have parallel but unequal bases, so it is a counterexample.

  3. A necessary visible feature need not be a sufficient definition.

No. The bases must also be congruent polygons and the lateral edges parallel in the prism structure.

Checks and common pitfalls: A necessary visible feature need not be a sufficient definition.

Reasoning checklist · self / teacher assessment
  • State the relevant definition, condition or model.
  • Show a valid calculation, proof or counterexample.
  • Interpret the conclusion with its restrictions.

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

09 / Standard#Your turn

A sphere has radius r. What is the distance of every surface point from its centre?

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
Identify the defining faces, cross-sections and generating elements of the solid.
Hint 2
Use the defining locus of a sphere.
Worked solution

  1. Identify the defining faces, cross-sections and generating elements of the solid.

  2. Calculate or simplify this relation.

    ∣OP∣=r=3|OP|=r=3

  3. Interior points may be closer; the statement concerns the surface.

The requested value is 3.

Checks and common pitfalls: Interior points may be closer; the statement concerns the surface.

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

10 / Standard#Your turn

Verify Euler’s relation for an 6-gonal prism by giving V,E,F.

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

Working and explanation

BUILD THE REASONING

Hint 1
Identify the defining faces, cross-sections and generating elements of the solid.
Hint 2
Count two bases and n lateral faces.
Worked solution

  1. Identify the defining faces, cross-sections and generating elements of the solid.

  2. Calculate or simplify this relation.

    2n−3n+(n+2)=2(n=6)2n-3n+(n+2)=2\quad(n=6)

  3. This verifies the relation for this convex polyhedron family; it is not a proof for every polyhedron.

V=12, E=18, F=8; V−E+F=2.

Checks and common pitfalls: This verifies the relation for this convex polyhedron family; it is not a proof for every polyhedron.

Reasoning checklist · self / teacher assessment
  • State the relevant definition, condition or model.
  • Show a valid calculation, proof or counterexample.
  • Interpret the conclusion with its restrictions.

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

11 / Standard#Your turn

How many vertices does an 7-gonal prism have?

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

Working and explanation

BUILD THE REASONING

Hint 1
Identify the defining faces, cross-sections and generating elements of the solid.
Hint 2
Each base contributes n vertices.
Worked solution

  1. Identify the defining faces, cross-sections and generating elements of the solid.

  2. Calculate or simplify this relation.

    V=2n=2(7)=14V=2n=2(7)=14

  3. Corresponding base vertices are distinct and joined by lateral edges.

The requested value is 14.

Checks and common pitfalls: Corresponding base vertices are distinct and joined by lateral edges.

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

12 / Transfer#Your turn

How many edges does an 7-gonal pyramid have?

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

Working and explanation

BUILD THE REASONING

Hint 1
Identify the defining faces, cross-sections and generating elements of the solid.
Hint 2
Count base edges and apex-to-base edges separately.
Worked solution

  1. Identify the defining faces, cross-sections and generating elements of the solid.

  2. Calculate or simplify this relation.

    E=n+n=2(7)=14E=n+n=2(7)=14

  3. Do not count diagonals of the base as edges of the solid.

The requested value is 14.

Checks and common pitfalls: Do not count diagonals of the base as edges of the solid.

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

13 / Transfer#Your turn

Find the slant height of the right circular cone.

r=12,h=16r=12,\quad h=16
  • Use the domain, units and sampling assumptions stated in the question.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Identify the defining faces, cross-sections and generating elements of the solid.
Hint 2
The radius and vertical height form a right triangle.
Worked solution

  1. Identify the defining faces, cross-sections and generating elements of the solid.

  2. Calculate or simplify this relation.

    l=r2+h2=144+256=20l=\sqrt{r^2+h^2}=\sqrt{144+256}=20

  3. A slant height is not the perpendicular height.

The requested value is 20.

Checks and common pitfalls: A slant height is not the perpendicular height.

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

Focus on one question

End-of-lesson check and correction

Review your latest checked answers and explanations. A draft change requires a fresh check. Written work needs your self-assessment or a teacher’s review.

Enable JavaScript for a summary of your local work.

    Choose a foundation skill to revisit ↗

    Teacher preparation and assessment

    Question sequence

    • What must be true before using the main rule for basic solid figures?
    • 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 basic solid figures.
    • Defining structures: Prisms, pyramids, frustums and solids of revolution have different defining base and side relations.
    • Sections: Parallel sections of a pyramid are similar; metric ratios depend on distance from the apex.
    • Close with: conditions → representation → reasoning → check.

    Anticipated thinking

    • Expected reasoning: Prisms, pyramids, frustums and solids of revolution have different defining base and side relations.
    • Expected reasoning: Parallel sections of a pyramid are similar; metric ratios depend on distance from the apex.
    • Expected correction: Two parallel faces alone do not define a prism.

    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.

    No sign-in. Work stays in this browser. Export before clearing browser data. Written reasoning is assessed with a checklist.

    Curriculum and source notes ↗