Scale Drawings and Maps

What a scale says

A scale drawing shows something at a different size while keeping every proportion — so it is similar to the original rather than merely a picture of it.

The scale states the relationship:

$$1 \text{ cm} : 50 \text{ km}$$

Every centimeter on the drawing stands for $50$ kilometers in reality.

Scale	Read as
$1$ cm : $50$ km	one centimeter represents fifty kilometers
$1$ : $200$	the real thing is $200$ times the drawing
$1$ inch : $4$ feet	one inch represents four feet

A scale written with no units, like $1:200$, works in any unit. One centimeter on the plan is $200$ centimeters in reality; one inch is $200$ inches. That is what makes unitless scales convenient.

Drawing to actual, and back

Both directions are one proportion.

A map has a scale of $1$ cm : $50$ km. Two towns are $7$ cm apart. How far apart are they really?

$$\frac{1}{50} = \frac{7}{x} \quad\Rightarrow\quad x = 350 \text{ km}$$

The same map. Two cities are $375$ km apart. How far apart on the map?

$$\frac{1}{50} = \frac{d}{375} \quad\Rightarrow\quad 50d = 375 \quad\Rightarrow\quad d = 7.5 \text{ cm}$$

Going to actual multiplies; coming back divides. A sense check settles which you need: the real distance is always the larger number on a map.

Finding the scale factor

Given a drawing length and its actual length, the factor is their quotient:

$$k = \frac{\text{actual}}{\text{drawing}}$$

A model car is $12$ cm long and the real car is $4.8$ m.

Convert to one unit first — $4.8$ m is $480$ cm:

$$k = \frac{480}{12} = 40$$

The scale is $1:40$.

Converting units first is not optional. Dividing $4.8$ by $12$ gives $0.4$, which is not a scale factor for anything.

Why area scales by k²

This is the part that catches people out, and it is worth understanding rather than memorizing.

A floor plan at $1:100$. A room measures $4$ cm by $3$ cm on the plan.

	Plan	Actual
length	$4$ cm	$400$ cm
width	$3$ cm	$300$ cm
area	$12$ cm²	$120{,}000$ cm²

The lengths went up by $100$. The area went up by:

$$\frac{120{,}000}{12} = 10{,}000 = 100^2$$

An area is a length times a length. Both have been multiplied by $k$, so the product is multiplied by $k$ twice:

$$\text{area factor} = k \times k = k^2$$

The same argument gives $k^3$ for volume, since a volume is three lengths multiplied.

Quantity	Scales by
length	$k$
area	$k^2$
volume	$k^3$

So doubling every length of a model quadruples the material needed to cover it and multiplies its capacity by eight.

Worked examples

Common mistakes

Practice problems