Similar Solids

Two solids are similar if one is a scaled copy of the other — same shape, all corresponding lengths in the same ratio k.

The three scaling rules:

• lengths (edges, radii, heights) scale by k
• surface areas scale by k²
• volumes scale by k³

So doubling every dimension gives 4 times the surface area and 8 times the volume.

Working backwards is the common exam question. If two similar solids have volumes in the ratio 27:8, then k³ = 27/8, so k = 3/2, and the surface areas are in ratio 9:4.

The rule of thumb: to go from a volume ratio back to a length ratio, take the cube root. From an area ratio, take the square root.

This is why a scale model weighs so much less than the scale suggests, and why doubling a recipe's pan size does not double the cooking time.

Similar solids have congruent corresponding angles and all corresponding lengths in a constant ratio k. Under such a similarity, any length scales by k, any area by k² and any volume by k³ — the exponent being the measurement's dimension.

Consequently the ratios interconvert: k = √(A₁/A₂) = ∛(V₁/V₂), which is how a length ratio is recovered from area or volume data.

The physical consequences are substantial. Since strength depends on cross-sectional area (k²) while weight depends on volume (k³), the strength-to-weight ratio falls as 1/k — the square-cube law, which limits how large a structure or organism of a given design can be. It also explains why small animals lose heat quickly, having high surface-area-to-volume ratio.

Care is required with the word "similar": all spheres and all cubes are similar to one another, but two cylinders are similar only if their radius-to-height ratios match. Similarity is not implied by sharing a shape name.