Similarity.
Match the corresponding features, then choose a length, area or volume scale factor.
Explain triangle similarity · find corresponding lengths · use square and cube scale factors
Match corresponding sides before scaling
Similar shapes have the same shape even when their sizes differ. Corresponding angles match, and every corresponding length uses one common scale factor.
Identify which vertices and sides match before dividing lengths. Rotation or reflection can disguise those correspondences, so use angle positions and labels rather than left-versus-right appearance. A scale factor must always name its direction: from the first shape to the second or back again.
Similar shapes have equal corresponding angles and proportional corresponding lengths. Rotation or reflection does not prevent similarity.
A side of length 4 corresponds to 6. Find the side corresponding to 5.
Scale factor from small to large = 64 = 1.5. The required side is 5 × 1.5 = 7.5.
Follow equal angles to identify corresponding sides. Do not match sides merely because they are both horizontal in the drawing.
Pause and explainIf a 4 cm side matches a 10 cm side, what is the enlargement factor from the smaller shape?
10 ÷ 4 = 2.5. In the reverse direction it would be 4 ÷ 10 = 0.4.
Explain why triangles are similar
Two matching angles determine a triangle’s third angle because its angle sum is 180°. This is why angle information can establish triangle similarity without measuring every side.
State the equal angles and give their reasons, such as corresponding angles in parallel lines or a shared angle. Then write the triangles in matching vertex order. Once similarity is established, corresponding-side ratios can be used to find missing lengths.
Two corresponding angles are equal, so the third pair is equal too.
All three pairs of corresponding sides are in the same ratio.
Two pairs of corresponding sides are in the same ratio, and their included angles are equal.
Two triangles each contain a right angle and an angle of 35°. Explain their similarity.
Two corresponding angles are equal, so the triangles are similar by AA. The third angle in each is 55°.
Pause and explainWhy is merely saying “the triangles look similar” insufficient?
A drawing may not be to scale. Similarity needs matching angle or side information supported by the given facts.
Lengths scale by k, areas by k², volumes by k³
If each side of a square doubles, the new square contains four copies of the original area, not two. If each dimension of a cube doubles, its volume contains eight original cubes.
Area involves two lengths being multiplied, so a length factor k creates area factor k². Volume involves three, creating k³. When an area or volume factor is given first, use a square or cube root to recover the length factor.
Volume comparison applies to similar solids, using the same length scale factor.
volume of imagevolume of original = k³
Similar shapes have areas 64 cm² and 144 cm². A small-shape side is 10 cm. Find the matching large side.
Area factor = 14464 = 94. Length factor = √(94) = 32.
Similar solids have volumes in the ratio 27 : 125. Find the length and surface-area ratios.
Take cube roots for lengths: 3 : 5. Square the length ratio for surface areas: 9 : 25.
Pause and explainIf similar solids have length scale factor 3, what is their volume scale factor?
3³ = 27. Three dimensions each contribute a factor of 3.
Your turn. Choose the right power.
Mark corresponding sides before choosing a ratio. Check whether the requested quantity is a length, area or volume.
Keep the direction of the scale factor fixed throughout. If using an area or volume ratio to find a side, recover the length factor with a root before multiplying the known length.
Similar triangles have matching sides 8 cm and 12 cm. Find the large triangle side matching 10 cm on the small triangle.
Reveal worked answer
A similar solid has twice every corresponding length. What are its surface-area and volume factors?
Reveal worked answer
Two similar solids have volumes 64 cm³ and 216 cm³. Find the ratio of their lengths and surface areas.
Reveal worked answer
Triangles have side lengths 3, 5, 7 and 6, 10, 14. Explain why they are similar.