Angle reasoning, similarity and mensuration · Higher
| English | 中文 | Pinyin |
|---|---|---|
| scale factor/skeɪl ˈfæktə/ | 相似比 | xiāng sì bǐ |
Does volume scale like length?
- A model has lengths one third of the real object. How much smaller are its area and volume?
- This lesson studies scale factor 相似比: The multiplier that relates corresponding lengths in similar shapes.
Choose the mathematical structure
- For similar shapes with length scale factor k, areas scale by k² and volumes by k³. State angle reasons explicitly. A circle's tangent is perpendicular to the radius at the contact point.
- State the allowed inputs and units before calculating. An equation should express the relationship, not just record a calculator entry.
Which description correctly defines scale factor?
The multiplier that relates corresponding lengths in 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.
If model-to-real length factor is 3, a model area of 12 cm² gives 12×3²=108 cm² and a model volume of 8 cm³ gives 8×3³=216 cm³. A cylinder with r=3,h=5 has volume πr²h=45π.
Angle reasoning, similarity and mensuration
For similar shapes with length scale factor k, areas scale by k² and volumes by k³
Match the scale factors to the correct dimension before using 12 cm² or 8 cm³.
Two similar figures have length factor 3. Find the area factor.
Area scales by the square of the length factor: 3²=9.
Test a tempting shortcut
- Equal angles alone establish similarity, not equal size. Use corresponding lengths in the same order. Convert linear units before calculating area or volume, or square/cube the conversion factor correctly.
- When a shortcut fails, identify the assumption it breaks. Keep an exact value until the requested final rounding.
Doubling every length of a solid doubles its volume. This claim is false. Explain which definition or assumption it violates.
Two similar solids have length factor 3. Find the volume factor.
Volume scales by the cube: 3³=27.
Doubling every length of a solid doubles its volume.
Equal angles alone establish similarity, not equal size. Use corresponding lengths in the same order. Convert linear units before calculating area or volume, or square/cube the conversion factor correctly.
Interpret a new situation
- A geometric proof should name the relevant theorem, identify the equal angle or ratio, and draw the conclusion. A scale drawing is evidence only when the task permits measurement.
- A complete solution gives the mathematical result and explains what it means. Check that it is possible in the stated context.
Find the sum of interior angles of a pentagon in degrees.
Interior-angle sum=(5-2)×180=540°.
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 · 3.4. 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 multiplier that relates corresponding lengths in similar shapes. Choose the relationship, show the method, check its assumptions and interpret the result.