Similarity

• Similar figures: corresponding angles equal, corresponding lengths in the same ratio.
• Length scale factor k = (length on new shape) ÷ (matching length on old shape).
• Maps and scale drawings use the same idea with a fixed scale.
• Higher: area scale factor = k²; volume scale factor = k³.

Same angles, lengths in proportion

Two figures are similar if one is an enlargement of the other. Their angles are unchanged; every length is multiplied by the same scale factor k. Triangles are similar if their angles are equal — a line drawn parallel to one side of a triangle cuts off a smaller similar triangle. Redraw the two triangles separately, the same way round, before matching sides.

Using a scale

A map scale may be written as a ratio (1 : 25 000) or in words (1 cm represents 2 km). Change real distances into the same units as the map before dividing.

k, k² and k³

Area is length × length, so areas scale by k². Volume (and capacity, and mass of the same material) is length × length × length, so it scales by k³. To go backwards, square-root an area ratio or cube-root a volume ratio to get k.