E4.4 - Area and volume scale factors

Why areas scale with the square

If a model's lengths are doubled, does its volume double too? We will answer that after seeing what doubling does to area. A $4\,\mathrm{cm}$ by $3\,\mathrm{cm}$ rectangle has area $12\,\mathrm{cm}^2$. Doubling both lengths gives $8\times6=48\,\mathrm{cm}^2$: four times the area.

[DIAGRAM: asset_name: Two length factors make an area factor; asset_slug: similar_area_grid; caption: Doubling both directions fits four original rectangles into the enlarged rectangle.; description: A four-by-three rectangle is compared with an eight-by-six rectangle divided into four congruent four-by-three rectangles. Corresponding horizontal and vertical lengths are labelled. Pale blue and sage distinguish original and enlarged boundaries; all dimensions are in centimetres.]

If every length is multiplied by $k>0$, a rectangle's area becomes $(kl)(kw)=k^2lw$. A triangle works the same way: its base and perpendicular height both scale by $k$, while the factor one half stays unchanged. Dividing other polygons into triangles gives the same result. For circles, $\pi(kr)^2=k^2\pi r^2$.

Only one common length factor gives this rule. Increasing a rectangle's width alone by $3$ multiplies its area by $3$, not $9$; it generally changes the shape rather than making a similar enlargement.

Change only the scale factor to $0.6$. A factor below one makes both lengths smaller.

The same rule applies to the entire boundary of similar solids.

Why volumes scale with the cube

Volume has three length directions. A $2\,\mathrm{cm}$ cube has volume $8\,\mathrm{cm}^3$; doubling its edge gives a $4\,\mathrm{cm}$ cube of volume $64\,\mathrm{cm}^3$. Two layers of four original cubes fit inside it: eight copies, not two.

[DIAGRAM: asset_name: Three length factors make a volume factor; asset_slug: similar_volume_blocks; caption: Two copies along each of three directions make eight original cubes.; description: An exact perspective cube with edge two centimetres is compared with a cube of edge four centimetres divided by three central planes into eight equal edge-two cubes. Pale fills and dashed hidden subdivisions show two positions across, two back and two up.]

For a cuboid, scaling each dimension by $k$ gives $(kl)(kw)(kh)=k^3lwh$. For a prism, the end area scales by $k^2$ and the length by $k$, giving $k^3$. Cone and sphere formulas also contain three multiplied lengths, so they give the same rule for similar solids.