Length, area and volume scale factors · Higher
| English | 中文 | Pinyin |
|---|---|---|
| area scale factor/ˈeərɪə skeɪl ˈfæktə/ | 面积比例因子 | miàn jī bǐ lì yīn zi |
A model doubles every length. Each rectangular face doubles in two directions, while its volume doubles in three directions.
- A model doubles every length. Each rectangular face doubles in two directions, while its volume doubles in three directions.
- This lesson studies area scale factor 面积比例因子: The squared length factor between similar shapes.
Choose the mathematical structure
- For similar shapes with corresponding length factor k, area factor is k² and volume factor is k³. Name the direction of the comparison: model to real or real to model. Recover k from an area ratio by a square root and from a volume ratio by a cube root. A change in only one dimension does not create similar solids. Higher links corresponding sides to equal trigonometric ratios.
- State the allowed inputs and units before calculating. An equation should express the relationship, not just record a calculator entry.
Which description correctly defines area scale factor?
The squared length factor between similar shapes.
Work through a checked case
- Check the result against the starting quantities. Substitute into the original relation, or compare the graph and numerical answer where appropriate.
Enlarging a box from 2×3×4 to 6×9×12 multiplies lengths by 3. Its volume changes from 24 to 648, a factor of 27; a 2×3 face changes area from 6 to 54, a factor of 9. Length ratio 2:5 corresponds to area ratio 4:25 and volume ratio 8:125. If similar shapes have areas 20 and 80 cm², the larger-to-smaller length factor is √(80/20)=2. If similar solids have volumes 16 and 128 cm³, k=∛8=2. Higher: triangles with the same acute angle have equal opposite/hypotenuse ratios; doubling both lengths leaves sinθ unchanged.
Length, area and volume scale factors
For similar shapes with corresponding length factor k, area factor is k² and volume factor is k³
Classify the worked-case statements, then explain the units or invariant that justifies each decision.
Find the area factor for length factor 3.
A face scales in two directions: 3²=9.
Test a tempting shortcut
- Areas do not scale by k, and volumes do not scale by k². Similarity needs every corresponding length to share the same factor. Reversing a ratio requires the reciprocal factor.
- When a shortcut fails, identify the assumption it breaks. Keep an exact value until the requested final rounding.
Doubling all lengths of a solid doubles its volume. This claim is false. Explain which definition or assumption it violates.
Find the volume factor for length factor 3.
A solid scales in three directions: 3³=27.
Doubling all lengths of a solid doubles its volume.
Areas do not scale by k, and volumes do not scale by k². Similarity needs every corresponding length to share the same factor. Reversing a ratio requires the reciprocal factor.
Interpret a new situation
- AQA R12 Foundation includes ratios and scale factors for lengths, areas and volumes. Higher adds links to similarity including trigonometric ratios. Check dimensions and compare a simple box or rectangle before applying the general factor.
- A complete solution gives the mathematical result and explains what it means. Check that it is possible in the stated context.
Find the length factor when similar areas have ratio 20:80, smaller to larger.
The area factor is 80/20=4; its square root is 2.
Match each part of a complete solution to its purpose.
An assumption justifies the model; a check tests the result; interpretation connects it to the question.
Use this in your course
- 8300 · Higher · 3.3. Match the target tier and specification before assigning extensions.
- Give the method before the final answer, and use the paper's calculator and formula rules. Review a wrong answer by locating the first invalid step.
The squared length factor between similar shapes. Choose the relationship, show the method, check its assumptions and interpret the result.