Perimeter measures the total length of a shape's boundary, area measures the flat space that shape covers, and volume measures the space a three-dimensional solid actually occupies. These three measurements sound like they should simply scale together proportionally when a shape gets bigger, but they genuinely don't: scaling a shape up by a given factor multiplies its perimeter by that same factor, but multiplies its area by the factor squared, and multiplies its volume by the factor cubed.
Perimeter is one-dimensional, so it scales directly with the scaling factor
Perimeter is fundamentally a length measurement, a one-dimensional quantity, so doubling every one of a shape's linear dimensions simply doubles its perimeter directly, with no extra multiplication involved, since each individual side or boundary segment just gets twice as long. This direct, one-to-one scaling is exactly what makes perimeter behave the way most people's intuition expects size to behave in general, which is precisely why it's such a genuine surprise when area and volume turn out not to follow that same simple pattern.
Area and volume are two- and three-dimensional, so they scale by the factor raised to a power
Area is a two-dimensional quantity, built from two linear dimensions multiplied together, so doubling every linear dimension doubles area twice over, once for each of the two dimensions involved, giving a total area increase of four times, the scaling factor squared. Volume is a three-dimensional quantity, built from three linear dimensions multiplied together, so doubling every linear dimension multiplies volume by two three separate times, giving a total volume increase of eight times, the scaling factor cubed. This is exactly why a shape scaled up modestly in every linear dimension can still hold dramatically, disproportionately more inside it, a pattern with genuinely real consequences for everything from packaging design to how an animal's body actually has to scale as it grows larger.
What we're still unsure about
That perimeter scales linearly while area scales with the square and volume with the cube of a shape's linear scaling factor is well established, rigorously proven geometry confirmed for every regular and irregular shape alike. What's more genuinely a matter of ongoing practical application is exactly how significant this scaling mismatch actually turns out to be for a given real-world design or biological question, since the effect's real-world consequences, why larger animals need proportionally thicker bones, why larger containers are often more material-efficient per unit volume, depend on the specific details of the situation, and researchers and engineers across biology and design continue to apply this same underlying scaling principle to genuinely different real-world problems.
This sits inside Area, Perimeter & Volume, one of eight topics in Geometry, one of seven domains in Mathematics, one of seventeen subjects the app can quiz you on.