Map scales for distance and area

A scale \(1:k\) means one map length represents \(k\) real lengths in the same unit. Area scales by \(k^2\), because both length and width are enlarged.

Example

At scale \(1:50000\), \(1\text{ cm}\) represents \(50000\text{ cm}=0.5\text{ km}\). A map distance of \(6.4\text{ cm}\) represents \(3.2\text{ km}\).

Each \(1\text{ cm}^2\) represents \((0.5\text{ km})^2=0.25\text{ km}^2\). A region of \(12\text{ cm}^2\) on the map represents \(3\text{ km}^2\).

Do not apply a length multiplier to an area. State the units at each stage; converting \(\text{cm}^2\) to \(\text{km}^2\) also requires a squared conversion.