Scale Factors and Maps

Every length must be changed by the same multiplier. This ensures that the drawing and the real object are similar: corresponding angles remain equal and corresponding lengths have a constant ratio. Scaling one length differently would distort the representation.

Understanding a scale factor

The scale factor from an original length to a new length is

A scale factor less than 1 produces a reduction, while a scale factor greater than 1 produces an enlargement. The direction matters: the scale factor from the drawing to reality is the reciprocal of the scale factor from reality to the drawing.

Before finding a scale factor, express both lengths in the same unit. A comparison such as centimetres divided by metres does not yet give a valid scale factor.

Reading a ratio scale

A scale written as $1:100$ means that 1 unit on the drawing represents 100 of the same units in reality. It could mean 1 cm represents 100 cm, or 1 mm represents 100 mm. The ratio itself has no unit because the units on both sides must match.

For a scale of $1:n$:

and

For example, a wall that is 1 metre long is 100 cm long. At a scale of $1:100$, its drawing length is therefore 1 cm. At a scale of $1:500$, the same wall is drawn as $0.2$ cm, which is 2 mm.

A general scale $a:b$ means that $a$ units on the representation correspond to $b$ units in reality. The multiplier from the drawing to reality is therefore $b/a$; the multiplier in the reverse direction is $a/b$.

Using a scale diagram

To find a real measurement from a scale diagram:

1. identify the scale;
2. measure or read the required length on the diagram;
3. make the units consistent;
4. multiply by the drawing-to-reality scale factor;
5. convert the result to a suitable real-world unit if needed.

To construct a scale diagram, reverse the process: convert each real length into a drawing length using the same scale. Use a ruler accurately and preserve the given angles and arrangement of the parts. A diagram that merely looks approximately correct is not necessarily drawn to scale.

Applying scales to maps

A map is a scale representation, so map distances and real distances are connected by the same proportional reasoning. A ratio scale such as $1:n$ compares distances in identical units. A statement scale gives the conversion directly in words.

For a straight-line distance, measure between the required points, apply the map scale and then give the real distance in an appropriate unit. When finding a map distance from a real distance, convert the real distance into compatible units first and divide by the scale multiplier.

The scale applies to lengths in every direction, not just horizontal distances. However, the calculated real distance can only be as accurate as the printed map and the measurement taken from it.

Comparing two drawing scales

The same object takes up different amounts of space at different scales. At $1:50$, each drawing unit represents 50 real units; at $1:100$, it represents 100 real units. The $1:100$ version therefore needs half the drawing length of the $1:50$ version.

More generally, when changing a drawing of the same object from scale $1:a$ to scale $1:b$,

Thus, changing from $1:500$ to $1:200$ multiplies every drawing length by $500/200=2.5$. The denominator is smaller, so the new drawing is larger. This provides a useful sense-check before calculating.