LEARN · EXPLAIN · REVISE

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.

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

TOPIC 01

Basic solid figures

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

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.

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

01 / Foundation#Your turn

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

Identify the defining faces, cross-sections and generating elements of the solid.
Calculate or simplify this relation.
$\lambda=1/2;\quad S_{\text{section}}/S_{\text{base}}=\lambda^2=1/4$
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.

Teacher preparation and assessment

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.

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.