Why does doubling the size need four times the material?

Because area is two dimensional. Each of the two dimensions doubles, so the area is two times two. Volume adds a third dimension and goes to eight.

Math & Statisticsgeometry

Scale Factor Calculator

Scale factor with its area and volume consequences. Doubling a length quadruples the area and multiplies volume by eight, which is why scaling a recipe or a structure is never proportional.

Also called: scale calculator, square cube law.

Scale factor

1.5

1.5x on length. Area scales by 2.25x and volume by 3.38x, because area goes as the square and volume as the cube.

Area multiplies by

2.25

Volume multiplies by

3.38

Change in length

50%

New area

129,600

Factor to go back

0.67

How this is calculated

Method and background

How this is calculated

The scale factor is the ratio of lengths. Area involves two dimensions and so scales by the square; volume involves three and scales by the cube. This square-cube law is why a photograph enlarged fifty percent needs 2.25 times the ink, why a scaled-up animal cannot support its own weight, and why doubling a cake tin diameter needs four times the batter rather than two.

the square-cube law: doubling length quadruples area and multiplies volume eightfold

k: Scale factor

Worked examples

Each of these is asserted on every build. If a change to the engine ever moved one of these answers, the build would fail before the page could print it.

a fifty percent enlargement

Original dimension: 240
New dimension: 360
Original area: 57,600

Scale factor1.5

1.5 squared and cubed

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doubling quadruples the area

Original dimension: 100
New dimension: 200
Original area: 10,000

Scale factor2

boundary: the definitional case

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Method and limits

What it assumes

Uniform scaling in every dimension.

What it deliberately does not model

Non-uniform scaling in one axis breaks the square and cube relationships.

Formula version 1.0.0 · definition 1.0.0 · United States · Report a problem with this calculator

Frequently asked questions

Why does doubling the size need four times the material?: Because area is two dimensional. Each of the two dimensions doubles, so the area is two times two. Volume adds a third dimension and goes to eight.