Angle reasoning, similarity and mensuration
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.
The method
- 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.
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.
A worked case
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.
Where it goes wrong
- 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.
Not true: Doubling every length of a solid doubles its volume. Volume scales by the cube of the length factor: doubling every length multiplies the volume by 8.
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.
Volume scales by the cube of the length factor.
難易度 1/5 · 100 コイン + 今日初回完了時に小ボーナス付与
True. Volume scales by the cube of the length factor: doubling every length multiplies the volume by 8.
Applying it
- 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.
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.
In your course
- Applications and Interpretation HL, the course first assessed in 2021.
- Every paper allows a calculator, but method marks still need the expression or equation written before its value.
- Scale factor: The multiplier that relates corresponding lengths in similar shapes.
- For similar shapes with length scale factor k, areas scale by k² and volumes by k³.
- Watch for: equal angles alone establish similarity, not equal size.
Write from a volume factor of 216
- A length scale factor is 6.
- Find the area factor and the volume factor. Doubling lengths does not double the volume.
Lengths are enlarged by a scale factor of 6. What are the area and volume factors, and why does doubling a length not double the volume?
Area scales by the square of the length factor. Volume scales by the cube.
What is the volume factor when the length factor is 6?
難易度 1/5 · 100 コイン + 今日初回完了時に小ボーナス付与
6 cubed is 216. The area factor is 36. Other options — 36 (the area factor); 27 (the lesson volume factor for length factor 3).