Cube a Common Length Scale Factor
If all corresponding lengths change by a factor k, base areas change by k² and heights by k.
About 20 minutes, plus any extended practice
Supplied materials
- Paper or a browser scratch area; a calculator is optional unless rounding is requested.
- Supplied teaching case: A solid has volume 10 cm³. All corresponding lengths are multiplied by 3. Find the similar solid’s volume. Candidate answers: (A) 13 cm³ (B) 270 cm³ (C) 30 cm³ (D) 90 cm³.
- Independent check inputs: A solid has volume 15 cm³. All corresponding lengths are multiplied by 3. Find the similar solid’s volume. Candidate answers: (A) 135 cm³ (B) 18 cm³ (C) 405 cm³ (D) 45 cm³.
Learn the idea
Three-dimensional formulas relate solid shapes to base areas, heights and radii. Similar solids change every corresponding length together, making volume respond to the cube of a scale factor rather than its first power. If all corresponding lengths change by a factor k, base areas change by k² and heights by k. Their product makes the similar solid’s volume change by k³.
Here is the method on the supplied teaching case. Three independent dimensions each scale by 3, so volume scales by 3³=27. Multiply 10 by 27 to obtain 270 cm³. Area would instead scale by 9.
The cube rule needs similarity in all dimensions. Scaling only height or only radius creates a different factor; verify that every corresponding length is changed before applying a volume scale.
- Write or sketch the supplied teaching case, labeling its quantities and restrictions.
- Apply the method shown in the worked solution, keeping each intermediate equality or justified inference.
- Solve the independent check with its changed inputs before revealing feedback.
- Complete the three practice cases; explain which mathematical rule rules out one distractor in each case.
See the method in action
Teaching case: A solid has volume 10 cm³. All corresponding lengths are multiplied by 3. Find the similar solid’s volume. Candidate answers: (A) 13 cm³ (B) 270 cm³ (C) 30 cm³ (D) 90 cm³. Solution: Three independent dimensions each scale by 3, so volume scales by 3³=27. Multiply 10 by 27 to obtain 270 cm³. Area would instead scale by 9. The final answer is 270 cm³. The cube rule needs similarity in all dimensions. Scaling only height or only radius creates a different factor; verify that every corresponding length is changed before applying a volume scale.
Check your reasoning
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Helpful tips
- If all corresponding lengths change by a factor k, base areas change by k² and heights by k.
- The cube rule needs similarity in all dimensions.
See the answer and explanation
405 cm³
Three independent dimensions each scale by 3, so volume scales by 3³=27. Multiply 15 by 27 to obtain 405 cm³. Area would instead scale by 9.
Take it further
Changed-input task: A solid has volume 45 cm³. All corresponding lengths are multiplied by 3. Find the similar solid’s volume. Candidate answers: (A) 1215 cm³ (B) 135 cm³ (C) 405 cm³ (D) 48 cm³. Show the mathematical reason, not only the selected answer. Expected result: 1215 cm³. Three independent dimensions each scale by 3, so volume scales by 3³=27. Multiply 45 by 27 to obtain 1215 cm³. Area would instead scale by 9.
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