Scaling: double the lengths

Make every length of a box twice as long and it needs 4 times the card but holds 8 times as much.

Twice as long does not mean twice as much card

Here is a belief that feels like common sense: make a box twice as big in every direction and it needs twice the card and holds twice as much. Both halves of it are wrong. Take a 1 cm square and double its sides. The new square is 2 cm by 2 cm, and four of the old squares fit inside it. Triple the sides and nine fit. Area is length times length, so when every length is multiplied by a scale factor k, area is multiplied by k × k = k².

Squares are not special. A circle with twice the radius has four times the area, and a photo printed twice as wide and twice as tall uses four times the paper.

Volume grows faster still

Volume is length times length times length, so it grows by k³. Double a 1 cm cube and you get a 2 cm cube with room for 2 × 2 × 2 = 8 of the small ones. In numbers, a cube with 2 cm edges has surface 24 cm² and volume 8 cm³. With 4 cm edges it has 96 cm² and 64 cm³, which is 4 times and 8 times as much. A made-up parcel box shows the same thing. A box 20 cm × 15 cm × 10 cm uses 1,300 cm² of card and holds 3,000 cm³, or 3 litres. The box 40 cm × 30 cm × 20 cm uses 5,200 cm² (four times as much) and holds 24,000 cm³, or 24 litres (eight times as much). Press Double lengths in the model on the box, the cylinder, the cone and the sphere. Whatever the shape, the readout shows area × 4 and volume × 8.

Only when every length changes together

The other half of the trap is the opposite mistake: using k² or k³ when only one length changed. Double the height of a cylinder but keep its radius, and the volume doubles. Nothing more. Solids whose lengths are all in the same proportion are called similar. A small and a large tin of the same design, a toy car and a real car, or a model building and the real building are similar if the model was made to scale.

A 1:50 model and a tin of paint

Architects build models at scales like 1 to 50. Every length in the model is 1/50 of the real one, so every area in it is 1/2,500 of the real area and every volume is 1/125,000 of the real volume. A model room that holds 0.4 litres of air stands for a real room of 50,000 litres, which is 50 m³. Paint makes the point clearly. However little paint the model takes, the real building needs 2,500 times as much, not 50 times.

Animals follow the same arithmetic. Heat is made by the whole body, a volume, and escapes through the skin, an area. For a cube with edge a, surface ÷ volume = 6a² ÷ a³ = 6/a. A 1 cm cube has 6 cm² of surface for every cm³, but a 6 cm cube has only 1. Large animals have less skin for each kilogram of body, which helps them keep warm, and small animals lose heat quickly. The same calculation explains why crushed ice cools a drink faster than one big cube.