If the length, width, and height of a rectangular solid box were each increased by 50%, by what percentage would the volume be increased?

It's a common trick of the mind to assume that if you increase three dimensions by 50% each, the total volume would just increase by the sum, or 150%. However, because volume is a three-dimensional measurement, the increases are compounded, leading to a much more dramatic result. Each dimension doesn't just add its increase; it multiplies it across the others, causing the volume to balloon exponentially rather than grow in a straight line.

Let's break down the math. An increase of 50% means the new dimension is 150% of the old one, or 1.5 times as large. Since volume is length × width × height, the new volume becomes (1.5 × L) × (1.5 × W) × (1.5 × H). When you multiply the numbers together, you find that 1.5 × 1.5 × 1.5 equals 3.375. This means the new box is 3.375 times the size of the original. To express this as a percentage, the new volume is 337.5% of the original. The increase is this new total minus the original 100%, which gives us the final 237.5%.

This concept is a practical demonstration of the square-cube law, a principle identified by Galileo. It states that as an object grows, its volume (which is cubed) increases faster than its surface area (which is squared). This law has massive implications in the real world, explaining why giant insects from science fiction couldn't support their own weight and why larger animals have a harder time cooling down than smaller ones.