2.1.10 Conversion between Distance → Area, Volume

Students find linear conversions relatively simple (for example 1 ft = 12 in), however when asked to find how many cubic inches are in cubic feet, they want to revert back to the linear conversion, which is incorrect (1 ft³ ≠ 12 in³). When converting between linear distance to areas and volumes you must square or cube the conversion factor, respectively. So in our example, we know that:

$1 \text{ ft} = 12 \text{ in} \Rightarrow 1 \text{ ft}^3 = (12)^3 \text{ in}^3 = 1728 \text{ in}^3$

Another example converting ft² to yd²:

$1 \text{ yd} = 3 \text{ ft} \Rightarrow 1 \text{ yd}^2 = (3)^2 \text{ ft}^2 = 9 \text{ ft}^2$

Problem Set 2.1.10