Angle reasoning, similarity and mensuration · Extension
הוצג| English | משמעות |
|---|---|
| scale factor/skeɪl ˈfæktə/ | גורם קנה מידה |
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?
רמת קושי 1/5 · 100 מטבעות + בונוס קטן בהשלמה הראשונה היום
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.
רמת קושי 2/5 · 140 מטבעות + בונוס קטן בהשלמה הראשונה היום
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.
רמת קושי 2/5 · 140 מטבעות + בונוס קטן בהשלמה הראשונה היום
Volume scales by the cube: 3³=27.
Volume scales by the cube of the length factor.
רמת קושי 1/5 · 100 מטבעות + בונוס קטן בהשלמה הראשונה היום
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.
רמת קושי 2/5 · 140 מטבעות + בונוס קטן בהשלמה הראשונה היום
Interior-angle sum=(5-2)×180=540°.
Match each scale factor to what it multiplies.
רמת קושי 2/5 · 140 מטבעות + בונוס קטן בהשלמה הראשונה היום
Area is a two-dimensional measure and volume three-dimensional, so the length factor is applied twice or three times.
In your course
- OxfordAQA International GCSE Mathematics (9260), Extension tier, specification section 3.3.
- 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 real area of 80
- A model uses length scale factor 4. A model area is 5 square centimetres.
- Find the real area and the volume scale factor.
The length scale factor from model to real object is 4. A model area is 5 square centimetres and a model volume is 2 cubic centimetres. What are the real area and the volume scale factor?
4 squared is 16 and 5 times 16 is 80. 4 cubed is 64.
What is the real area?
רמת קושי 1/5 · 100 מטבעות + בונוס קטן בהשלמה הראשונה היום
The area factor is 16, and 5 times 16 is 80.