Sometimes scales come with units, and sometimes they don’t. For example, a map of Nebraska may have a scale of 1 mm to 1 km. This means that each millimeter of distance on the map represents 1 kilometer of distance in Nebraska. The same scale without units is 1:1,000,000, which means that each unit of distance on the map represents 1,000,000 units of distance in Nebraska. This is true for any choice of unit.
To see that these two scales are equivalent, notice there are 1,000 millimeters in 1 meter and 1,000 meters in 1 kilometer. This means there are $1,\!000 \boldcdot 1,\!000$ or 1,000,000 millimeters in 1 kilometer. So the actual distances in Nebraska are 1,000,000 times as far as the distances on the map.
A scale tells us how a length on a drawing corresponds to an actual length, and it also tells us how an area on a drawing corresponds to an actual area.
For example, if 1 centimeter on a scale drawing represents 2 meters in actual distance, what does 1 square centimeter on the drawing represent in actual area? The square on the left shows a square with side lengths 1 cm, so its area is 1 square cm.
The square on the right shows the actual dimensions represented by the square on the left. Because each side length in the actual square is 2 m, the actual square has an area of $2^2$ or 4 square meters.
We can use this relationship to find the actual area of any region represented on this drawing. If a room has an area of 18 cm2 on the drawing, we know that it has an actual area of $18 \boldcdot 4 = 72$ or 72 m2.
In general, if 1 unit on the drawing represents $n$ actual units, then one square unit on the drawing represents $n^2$ actual square units.