Geometry Lab reference
Similarity & scale factors
Why doubling the size changes more than you might expect.
Length × k · Area × k² · Volume × k³
The idea
Similar shapes have equal corresponding angles and proportional corresponding lengths. Under a positive uniform scale factor k, lengths multiply by k, areas by k², and volumes by k³. This distinction explains why a bigger package can hold much more while using proportionally less surface material.
Work through an example
Scale a 3 cm cube by a factor of 2. Its new edge is 6 cm. Surface area grows from 54 to 216 cm², a factor of 4. Volume grows from 27 to 216 cm³, a factor of 8.
What to watch for
The rule assumes uniform scaling in every dimension. Doubling only the height of a cylinder doubles volume; it does not multiply it by eight.
Make the formula move.
Explore this idea with real inputs.