Why Doubling the Size of a Shape Multiplies Its Volume by Eight

Double every length of a cube and its volume does not double. It grows eight times. This surprises many people, but it follows from one simple rule that applies to every shape, and it explains everything from why a larger pizza is a better deal to why elephants have thick legs.

The rule

If you multiply every length of a shape by a factor k, then every length grows by k, every area grows by k², and every volume grows by k³. Lengths are one-dimensional, areas are two-dimensional and volumes are three-dimensional, so each picks up one more factor of k.

A cube and a sphere

A cube with a 2 cm side has a volume of 8 cm³ and a surface area of 24 cm². Double the side to 4 cm and the volume is 64 cm³ (eight times as much) while the surface area is 96 cm² (four times as much). The same happens to a sphere: a radius of 5 cm gives a volume of 523.6 cm³ and a surface area of 314.2 cm², while a radius of 10 cm gives 4,188.8 cm³ and 1,256.6 cm². The volume ratio is 8 and the area ratio is 4.

Where this matters

• Pizza and pans. A 16-inch pizza has four times the area of an 8-inch pizza, not twice. A pan twice as wide and twice as deep holds eight times as much.
• Paint and material. Paint needed scales with area, but the weight of the object scales with volume. Doubling a statue’s size needs four times the paint and eight times the material.
• Model making. A 1:10 model has one tenth the length, one hundredth the area and one thousandth the volume of the real thing.
• Heat loss. Small objects have more surface for their volume, so they cool and dry faster. This is why a small potato bakes sooner than a large one.
• Strength. Galileo noticed in the 1600s that weight grows with volume (k³) while the strength of a limb grows only with its cross-section (k²), so large animals need proportionally thicker bones.

Working backwards

To double the volume of a container you do not double its dimensions. You multiply them by the cube root of two, about 1.26. In general, to change a volume by a factor F, scale the lengths by F to the power one third. To halve the amount of paint (area), scale by about 0.71, the square root of one half.

Try it on a real shape

In the 3D Shape Calculator, open More options and type a number in the box that multiplies every dimension. The results show the new volume and surface area beside the originals, for all 24 shapes. The Sphere Volume Calculator and the Cylinder Volume Calculator are good places to experiment.

Frequently asked questions

Why does doubling the size multiply the volume by eight? Volume has three dimensions: length, width and height. Doubling all three multiplies the volume by 2 × 2 × 2 = 8.

What happens to the surface area when you double the size? The surface area is two-dimensional, so it grows by 2 × 2 = 4 times.

How do I double the volume of a container? Multiply every dimension by the cube root of 2, about 1.26. For example, a 10 cm cube becomes a 12.6 cm cube.

Does this work for every shape? Yes, as long as every length is multiplied by the same factor so the shape stays the same. It does not apply if you stretch only one dimension.