AI Tutor
If a restaurant sells an 8-inch personal pizza for 8anda16−inchpizzafor8 and a 16-inch pizza for 16, which one gives you more food for your dollar?
Doubling the diameter actually quadruples the total pizza, because scaling a shape's lengths affects perimeter and area at fundamentally different rates.
What is the exact algebraic rule behind this sudden jump in size?

Linear Scaling and Perimeter

A scale factor kk is the constant multiplier applied to every linear dimension—such as length, width, radius, or slant height—of a geometric figure.
Because perimeter is a 1-dimensional sum of boundary lengths, multiplying each side by kk factors out directly: Pnew=k⋅PoldP_{\text{new}} = k \cdot P_{\text{old}}.
If the boundary only grows by kk, why does the interior space grow so much faster?

Area Scales by the Square of kk

An area measures two-dimensional coverage by multiplying two perpendicular linear dimensions together.
When both dimensions scale by kk, their product scales by k×kk \times k, meaning the new area is multiplied by k2k^2.