Mr Kidman's Classroom
Revise the key ideas
Similar figures and scale factors
- Similar figures have equal corresponding (matching) angles and proportional corresponding lengths. Congruent figures are the same size and shape, so their length scale factor is 1.
- Identify corresponding sides using matching angles and position, rather than choosing sides simply because they look similar on the diagram.
- Length scale factor from a smaller figure to a larger figure is . All corresponding lengths use the same factor.
Two similar right-angled triangles - Worked example: A side of 4 cm corresponds to 10 cm, so scale factor is 2.5. A corresponding 6 cm side becomes 15 cm; reversing the enlargement uses factor 0.4.
- For similar triangles, equal angles establish the same shape (AA similarity: two matching angles are equal). This is not a congruence test: equal angles alone do not force equal side lengths.
- Trigonometric ratios stay constant in similar right-angled triangles because corresponding sides scale together. This is why sin θ, cos θ and tan θ depend on the angle.
- Perimeters scale by the length factor. If each length triples, a perimeter of 18 cm becomes 54 cm.
Higher — area and volume scaling
- When the length factor is k, the area factor is k2 and the volume factor for similar solids is k3. Different dimensions require different powers.
- Worked example: Similar shapes with length ratio 2 : 5 have area ratio 4 : 25. An area of 12 cm2 on the smaller shape corresponds to 12 × = 75 cm2.
- Worked example: Similar solids have length factor 3. Their volumes have factor 27, so a volume of 20 cm3 becomes 540 cm3.
- To recover the length factor from an area factor, take a square root. An area factor 49 gives a length factor 7. From a volume factor 64, take a cube root to get 4.
- If similar containers have volume ratio 8 : 27, their height ratio is 2 : 3 and their surface-area ratio is 4 : 9.
- Mass scales like volume only if density is unchanged. Paint needed for a surface usually scales like area, whereas material filling a solid scales like volume.
This follows from mass = density × volume. If the material changes and its density changes, the mass ratio need not equal the volume ratio.
- Similarity must be established before applying square or cube scale rules. Changing just one dimension of a cuboid does not create a similar solid.
