<P> One can view the Euclidean plane as the complex plane, that is, as a 2 - dimensional space over the reals . The 2D similarity transformations can then be expressed in terms of complex arithmetic and are given by f (z) = az + b (direct similitudes) and f (z) = az + b (opposite similitudes), where a and b are complex numbers, a ≠ 0 . When a = 1, these similarities are isometries . </P> <P> The ratio between the areas of similar figures is equal to the square of the ratio of corresponding lengths of those figures (for example, when the side of a square or the radius of a circle is multiplied by three, its area is multiplied by nine--i.e. by three squared). The altitudes of similar triangles are in the same ratio as corresponding sides . If a triangle has a side of length b and an altitude drawn to that side of length h then a similar triangle with corresponding side of length kb will have an altitude drawn to that side of length kh . The area of the first triangle is, A = 1 / 2bh, while the area of the similar triangle will be A ′ = 1 / 2 (kb) (kh) = k A. Similar figures which can be decomposed into similar triangles will have areas related in the same way . The relationship holds for figures that are not rectifiable as well . </P> <P> The ratio between the volumes of similar figures is equal to the cube of the ratio of corresponding lengths of those figures (for example, when the edge of a cube or the radius of a sphere is multiplied by three, its volume is multiplied by 27--i.e. by three cubed). </P> <P> Galileo's square--cube law concerns similar solids . If the ratio of similitude (ratio of corresponding sides) between the solids is k, then the ratio of surface areas of the solids will be k, while the ratio of volumes will be k . </P>

Which is not a criterion for similarity of triangles