Similar Polygons

Similar figures have the same shape but not necessarily the same size. Two conditions must both hold: corresponding angles are congruent, and corresponding sides are in a constant ratio.

That ratio is the scale factor, k. Find it by dividing any pair of corresponding sides.

Congruence is the special case where k = 1.

The consequences for measurement are worth over-learning:

• lengths, including perimeter, scale by k
• areas scale by k²

So a figure scaled up by 3 has 3 times the perimeter but 9 times the area. Doubling a photo's dimensions quadruples the paper it needs.

As with congruence, the order of letters in a similarity statement records which vertices correspond.

Two polygons are similar when there is a vertex correspondence with all corresponding angles congruent and all corresponding sides proportional. Both conditions are required for polygons in general — a rectangle and a square have congruent angles but are not similar, while a square and a rhombus have proportional sides but are not similar.

Triangles are the exception where either condition alone suffices, which is what makes the triangle similarity criteria so powerful.

Equivalently, in transformational terms, two figures are similar when a sequence of rigid motions followed by a dilation maps one onto the other. The dilation contributes the scale factor k.

Under a similarity of ratio k, every length scales by k, every area by k² and every volume by k³ — the exponent being the dimension of the measurement. Angles are unchanged, since dilation preserves angle measure.

The distinction between the scale factor and its square is the most common source of error, and stating which quantity is being scaled before computing prevents nearly all of them.