When sizes change

Scale every length by k: areas scale by k², volumes by k³. Double every length and the area becomes 4 times, the volume 8 times.

Percentages:

• radius +10%: volume × 1.1³ = 1.331, an increase of 33.1%;
• volume +72.8% means × 1.728 = 1.2³, so the radius grew by 20%.

Adding 1 to one length:

• a cube a → a + 1 needs 3a² + 3a + 1 more unit cubes;
• a cuboid's length l → l + 1 adds wh;
• a cylinder's radius r → r + 1 adds π(2r + 1)h = 2πrh + πh.

For a cube, area and volume are linked: S = 6V^(2/3) and V = (S/6)^(3/2).

A famous surprise. A string 1 m longer than the Earth's equator stands 1/(2π) m ≈ 16 cm above it all the way round. And the same is true for the Moon, Jupiter or a volleyball, because 2π(R + h) – 2πR = 2πh: the R cancels.