Angle reasoning, similarity and mensuration · Higher
導入済み| English | 意味 |
|---|---|
| scale factor/skeɪl ˈfæktə/ | 相似比 |
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.
Two similar solids have length scale factor 3. Why is the volume factor 27?
難易度 1/5 · 100 コイン + 今日初回完了時に小ボーナス付与
Each of the three dimensions multiplies by 3, so volume multiplies by 3×3×3=27. Area uses the square and volume the cube of the length factor.
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.
難易度 2/5 · 140 コイン + 今日初回完了時に小ボーナス付与
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.
難易度 2/5 · 140 コイン + 今日初回完了時に小ボーナス付与
Volume scales by the cube: 3³=27.
Doubling every length of a solid doubles its volume.
難易度 1/5 · 100 コイン + 今日初回完了時に小ボーナス付与
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.
難易度 2/5 · 140 コイン + 今日初回完了時に小ボーナス付与
Interior-angle sum=(5-2)×180=540°.
Match each scale factor to what it multiplies.
難易度 2/5 · 140 コイン + 今日初回完了時に小ボーナス付与
Area is a two-dimensional measure and volume three-dimensional, so the length factor is applied twice or three times.
Use this in your course
- 4MA1 · Higher · 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.