Angles, lengths and area · Core
| 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
- Use angle facts with a stated reason. Similar shapes have equal corresponding angles and proportional corresponding lengths. Areas of rectangles and triangles come from their dimensions; compound shapes can be split into simpler parts.
- 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.
A rectangle of length 8 cm and width 5 cm has area A=LW=40 cm². A triangle on the same base and height has area A=bh/2=20 cm². For a pentagon, the interior-angle sum is (5-2)×180=540°.
Angles, lengths and area
Use angle facts with a stated reason
Explain why the triangle has half the rectangle area when base and height match.
Find the area of a rectangle of sides 8 cm and 5 cm.
Rectangle area=8×5=40 cm².
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.
A triangle and rectangle with the same base and height have the same area. This claim is false. Explain which definition or assumption it violates.
Find the area of a triangle of base 8 cm and height 5 cm.
Triangle area=8×5/2=20 cm².
A triangle and rectangle with the same base and height have the same area.
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
- Use a labelled sketch and appropriate units. This Foundation/Core lesson does not test area/volume scale factors or advanced circle-theorem proofs.
- 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
- 9260 · Core · 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 multiplier that relates corresponding lengths in similar shapes. Choose the relationship, show the method, check its assumptions and interpret the result.