Scale Drawings and Maps

A scale drawing or a map is a proportional representation of a real-world object or place. Because we cannot draw a city or a house at its actual size on a piece of paper, we shrink it down using a scale factor.

Understanding Map Scales

A scale tells you how the measurements on the drawing relate to the actual measurements in the real world. Scales can be written in two common ways:

• With units: e.g., $1\text{ cm} = 50\text{ km}$ (1 centimeter on the map represents 50 kilometers in reality).
• As a ratio: e.g., $1:500$ (1 unit on the map represents 500 of the same units in reality).

Finding Actual Distances

To find a real-world distance from a map, multiply the map distance by the scale factor.

Example: A map has a scale of $1:500$. A street measures $3\text{ cm}$ on the map. What is the actual length of the street?

1. The scale $1:500$ means $1\text{ cm}$ on the map equals $500\text{ cm}$ in real life.
2. Multiply the map distance by 500: $3\text{ cm} \times 500 = 1500\text{ cm}$
3. Convert to meters (since $100\text{ cm} = 1\text{ m}$): $1500\text{ cm} = 15\text{ m}$

The actual street is $15\text{ meters}$ long.

Creating a Scale Drawing

To create a drawing from real measurements, you divide the actual distance by the scale factor.

Example: You want to draw a map of a $200\text{ km}$ highway using a scale of $1\text{ cm} = 50\text{ km}$. How long should the line on your map be?

Divide the real distance by the scale value: $200\text{ km} \div 50\text{ km/cm} = 4\text{ cm}$

Your drawn line should be $4\text{ cm}$ long.

How Scales Affect Area

It is important to remember that scales apply to lengths (1D), but areas (2D) scale differently.

Rule: If the lengths of a drawing are scaled by a factor of $k$, the area is scaled by a factor of $k^2$.

Example: If a map's scale changes from $1:100$ to $1:200$, how does the drawn area change?

1. The new scale ($1:200$) means things are drawn half as large in length compared to the old scale ($1:100$). So, the length scale factor is $k = \frac{1}{2}$.
2. To find the effect on area, square the length scale factor: $k^2 = \left(\frac{1}{2}\right)^2 = \frac{1}{4}$

The new drawn area will be exactly $\frac{1}{4}$ the size of the original drawn area.