LEARN · EXPLAIN · REVISE

Find the volume of a pyramid frustum with parallel bases.

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

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.

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.

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

01 / Standard#Your turn

S_1=9,\quad S_2=36,\quad h=3

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

Choose the correct surface or volume formula and keep units and similarity powers consistent.
Calculate or simplify this relation.
$V=\frac h3(S_1+S_2+\sqrt{S_1S_2})=63$
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.

Teacher preparation and assessment

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?

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.

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.