9.04c - Area and volume scale factors

Why areas scale by the square

A square of side one contains one unit square. A square of side two contains four, so doubling a length does not merely double an area.

[DIAGRAM: asset_name: Counting squared scale factors; asset_slug: area-square-factor; caption: Doubling each perpendicular direction gives four times as many unit squares.; description: A one-by-one square and a two-by-two square are shown on equal-scale grids. Each unit cell is visible, with one and four unit-square counts. Corresponding horizontal and vertical lengths are labelled one and two, and the length factor two and area factor four are stated.]

For a rectangle of base $b$ and height $h$, an enlargement with positive length factor $k$ gives base $kb$ and height $kh$. Hence

A triangle behaves the same way: $\tfrac12(kb)(kh)=k^2(\tfrac12bh)$. Splitting polygons into triangles extends the result to their areas. Curved similar shapes obey the same relationship: for a circle, replacing $r$ by $kr$ gives $\pi(kr)^2=k^2\pi r^2$. Surface areas of similar solids also scale by $k^2$, because every corresponding face or surface patch scales in two directions.

The word “similar” guarantees that all corresponding lengths share the factor. Doubling only the width of a rectangle while keeping its height fixed doubles its area, but does not give a similar rectangle.

Change only the length factor to its reciprocal. A smaller image now has a smaller area factor.

Length ratios square in the same way: corresponding lengths $2:3$ give areas $4:9$. Keep the ratio direction consistent.

Use another fractional reduction.

Why volumes scale by the cube

A cube doubled in all directions contains four unit cubes in each horizontal layer and two layers: eight cubes altogether.

[DIAGRAM: asset_name: Counting cubed scale factors; asset_slug: volume-cube-factor; caption: Doubling width and depth gives four cubes per layer, and doubling height gives two layers.; description: A unit cube is compared with a two-by-two-by-two cube assembled from eight visible or indicated unit cubes. All three dimensions are labelled, with two layers of four described. The length factor is two and the volume factor is eight.]

For a cuboid with dimensions $a,b,c$, the scaled volume is

For any prism, the cross-sectional area scales by $k^2$ and its length scales by $k$, giving $k^2\times k=k^3$. The same three-direction relationship holds for other similar solids. Similarity includes every relevant dimension: cylinders with doubled radius but unchanged height are not similar cylinders.

Capacity follows internal volume when the internal shapes are similar. Do not assume that a stated outside length factor also scales the inside exactly if wall thickness or internal design changes.