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Why area and volume change faster than length

Scaling a length by k scales area by k² and volume by k³. This page derives both results rather than asserting them, and then draws the line between geometric volume and what a printer, a paint tin or a scale actually consumes.

Deriving k²: why area squares

Area is not a single length. It is a product of two independent lengths, and uniform scaling multiplies both of them by k. Take a rectangle:

original: 40 mm × 60 mm
scale factor: k = 1.5
new: (1.5 × 40) × (1.5 × 60) = 60 mm × 90 mm

Now compare the two areas:

original area = 40 × 60 = 2,400 mm²
new area = 60 × 90 = 5,400 mm²
5,400 ÷ 2,400 = 2.25 = 1.5²

The factor 2.25 is not a coincidence and not an empirical observation. Symbolically, the scaled area is (kL)(kW) = k²(LW), so the k appears once for the width and once for the height and the two collect into k². Any shape can be approximated by rectangles, so the result holds for circles, outlines, surfaces and irregular footprints too — a circle of radius kr has area π(kr)² = k²πr².

Deriving k³: why volume cubes

A solid has three independent lengths. Scale a box uniformly and every one of them picks up a factor of k:

(kL)(kW)(kH) = k³(L·W·H)

Numerically, with a 40 × 60 × 20 mm block at k = 1.5:

new dimensions: 60 × 90 × 30 mm
original volume = 40 × 60 × 20 = 48,000 mm³
new volume = 60 × 90 × 30 = 162,000 mm³
162,000 ÷ 48,000 = 3.375 = 1.5³

So a 50% increase in every length is a 125% increase in area and a 237.5% increase in volume. Nothing about the shape enters the derivation — only the count of independent lengths being multiplied.

Geometric volume is not material use

This is the distinction most often lost. k³ describes the geometric volume of uniformly scaled solid geometry. It does not automatically mean that filament usage, print time, weight, paint or cost scale by k³, and treating it as a material estimate is how a print budget goes wrong.

Several things break the relationship:

  • Wall thickness and shells. A hollow part with a fixed wall thickness does not gain interior material when it grows. Its shell volume tracks surface area, so it behaves closer to k² than k³. Wall thickness is set by the slicer in millimetres, not as a fraction of the model.
  • Infill. Infill is a percentage of interior volume, so it does follow volume — but only the interior, and only if the setting stays the same. Changing infill between prints changes the exponent you are effectively working with.
  • Supports. Support material depends on overhang geometry and orientation, not on the model volume, and often grows faster than the part when a feature crosses an angle threshold.
  • Orientation. Rotating a part changes layer count, support need and print time without changing volume at all.
  • Print time. Time is dominated by extrusion path length and layer count, which are area- and height-driven, so time and material rarely move by the same factor.
  • Density and process. Weight is volume × density, and density depends on material and process. Resin, sintered powder, cast metal and FDM with 15% infill give four different weights for identical geometry.
  • Cost. Cost includes machine time, failure rate, post-processing and minimum order quantities, none of which are geometric.

The honest formulation is: k³ is what the geometry does; material is what your process does with that geometry. If you need the second, model it in the 3D print scale calculator with your own wall, infill and orientation settings and treat the volume figure there as a geometric bound rather than a quote.

Counting it out

The exponent is a count, not a convention. Put the same figures on a 10 mm grid and the multiplication becomes something you can tally.

Counting how many times the factor 1.5 is applied in one, two and three dimensionsDrawn on a 10 mm grid at k = 1.5. In one dimension, 40 mm (4 units) becomes 60 mm (6 units). In two dimensions the column count goes 4 to 6 and the row count goes 6 to 9, so the cell count goes 24 to 54 and the area ratio is 54 divided by 24 = 2.25 = 1.5 squared. In three dimensions the 20 mm depth is 2 layers of 24 cells = 48 cells, and the 30 mm depth is 3 layers of 54 cells = 162 cells, so the volume ratio is 162 divided by 48 = 3.375 = 1.5 cubed. Each cell is 10 by 10 by 10 mm, that is 1,000 cubic mm, so the counts correspond to 48,000 and 162,000 cubic mm.ONE DIMENSION — 1.5 APPLIED ONCE40 mm = 4 units60 mm = 6 unitsTWO DIMENSIONS — 1.5 APPLIED TWICE60 mm90 mmcolumns 4 → 6rows 6 → 9cells 24 → 5454 ÷ 24 = 2.25 = 1.5²solid outline: originaldashed outline: scaledTHREE DIMENSIONS — 1.5 APPLIED THREE TIMES2 layers × 24 = 4840 × 60 × 20 mm×1.5 onall three3 layers × 54 = 16260 × 90 × 30 mm162 ÷ 48 = 3.375 = 1.5³cell = 1,000 mm³
Every cell is 10 mm, so the exponent is a count rather than a claim: 1.5 is applied once along a length, twice across an area (24 → 54 cells), and three times through a volume (48 → 162 cells).

Four columns become six and six rows become nine, so 24 cells become 54 — the factor 1.5 was applied along each of the two independent directions, and 54/24 = 2.25 = 1.5². Add depth and the two layers become three, so 48 cells become 162 and 162/48 = 3.375 = 1.5³. Each cell is 1,000 mm³, which is the same statement as 48,000 mm³ → 162,000 mm³.

The same thing on the way down: k = 0.75

Reductions are where the exponents are underestimated most often, because “a quarter smaller” sounds mild. Take the same block at k = 3/4:

length: 40 × 3/4 = 30 mm, 60 × 3/4 = 45 mm, 20 × 3/4 = 15 mm
area: 2,400 → 1,350 mm²  →  1,350/2,400 = 9/16 = 0.5625 = (3/4)²
volume: 48,000 → 20,250 mm³  →  20,250/48,000 = 27/64 = 0.421875 = (3/4)³

A 25% reduction in length removes 44% of the surface and 58% of the volume. That is the useful direction of the law when material is the concern, and the dangerous one when detail is: the wall that was 1.2 mm is now 0.9 mm, and nothing about the geometry warns you that a process minimum was crossed.

Consequence table

Every row below is computed with the same exact rational arithmetic the calculators use — k² and k³ are evaluated as fractions, not as rounded decimals multiplied together.

Factor kLengthArea (k²)Volume (k³)
0.5×50%25%12.5%
0.75×75%56.25%42.1875%
1×100%100%100%
1.25×125%156.25%195.3125%
1.5×150%225%337.5%
2×200%400%800%

Read the two ends of that table together. At k = 2 the volume is eight times the original; at k = 0.5 it is one eighth. A 25% increase in height nearly doubles volume, and a 28 mm miniature taken to 32 mm is only about 114% in length but roughly 49% more volume.

Where this bites

  • Paint and coatings. Coverage follows area, so a model at 150% needs more than twice the paint, not half again as much.
  • Paper and drawings. Doubling both sheet dimensions quadruples sheet area, which is exactly why ISO A-series sizes step by a factor of √2 per length instead.
  • Thin features on the way down. Wall thickness, pins, holes and embossed detail scale with everything else, and below the printable or castable minimum they stop existing. Reduction is where geometry silently becomes unmanufacturable.
  • Structure. Load capacity depends roughly on cross-sectional area (k²) while weight grows with volume (k³), so an enlarged structure is relatively weaker. This is the square-cube law, and it limits how far any design can simply be scaled up.

Three questions, not one

k, k² and k³ answer different questions, and only one of them is the question you typed. k is what the resize field wants. k² is how much surface exists now — paint, coating, sheet, cross-section. k³ is how much volume the geometry encloses, which is where a material estimate starts and not where it ends. Keeping the three apart is the whole practical content of the square-cube law; you can see them for your own figures in the scale factor calculator, and check the exponent you applied against the verification checks.