LEARN · EXPLAIN · REVISE

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.

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

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

Use volume conservation to find the linear scale first.
Calculate or simplify this relation.
$8r^3=R^3\Rightarrow r=R/2;\quad\frac{8r^2}{R^2}=2$
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.

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.