Dimensional ratios and similarity
When two shapes are similar, every linear measurement scales by the same factor k. But area scales by k² and volume by k³. OCR J560 Higher tests all three — and Foundation tests the basic linear scale factor.
Linear scale factor
If two similar shapes have a length scale factor k:
- Every length (side, perimeter, diagonal, height) scales by k.
E.g. a triangle with sides 3, 4, 5 cm scaled by k = 2 gives 6, 8, 10 cm.
Area scale factor = k²
If lengths scale by k, areas scale by k². So doubling lengths quadruples area.
E.g. similar rectangles with linear ratio 2 : 5 have area ratio 4 : 25.
Volume scale factor = k³
If lengths scale by k, volumes scale by k³.
E.g. similar cones with linear ratio 1 : 3 have volume ratio 1 : 27.
Working backwards from area or volume
If two similar cylinders have volumes 8 cm³ and 64 cm³:
- Volume ratio = 8 : 64 = 1 : 8.
- Linear ratio = ∛(1 : 8) = 1 : 2.
- Area ratio = (1 : 2)² = 1 : 4.
When are two shapes similar?
For triangles, ANY of these is sufficient:
- All three pairs of corresponding angles equal (AA similarity — two are enough since the third follows).
- All three pairs of corresponding sides in the same ratio.
- Two pairs of sides in the same ratio AND the included angle equal.
Setting up a ratio in a problem
Identify the corresponding sides — order them in the same direction (e.g. shortest to longest) in both shapes. Set up the equality:
larger side / smaller side = scale factor
then use it everywhere.
OCR mark scheme conventions
- M1 for stating the linear scale factor.
- M1 for raising it to the appropriate power for area (k²) or volume (k³).
- A1 for the final answer cao.
- Units must be square units for area, cubic units for volume.
⚠Common mistakes
- Using k for area instead of k².
- Using k² for volume instead of k³.
- Inverting the scale factor (smaller ÷ larger when bigger ÷ smaller was meant).
- Forgetting to take the cube root when given a volume ratio.
AI-generated · claude-opus-4-7 · v3-ocr-maths-leaves