Maps, Plans and Diagrams: Scale and Direction

Overview

Maps, plans and diagrams are scaled representations of real spaces. In Foundation Mathematics, you need to interpret these representations to find actual distances, areas, and directions — and to navigate in real-world contexts.

KEY TAKEAWAY: A scale tells you the relationship between a measurement on the diagram and the real-world measurement. Always apply the scale before drawing any conclusions about actual size.

Scale

A scale shows how a map measurement converts to a real measurement.

Written Scale

$$1\text{ cm} : 50\text{ m} \quad \text{(1 cm on map = 50 m in reality)}$$

Ratio Scale

$\$1 : 5000 \quad \text{(1 unit on map = 5000 of the same units in reality)}$$

Scale Bar

A drawn line on the map labelled with its real-world length.

Using a Scale to Find Real Distances

$$\text{Real distance} = \text{map distance} \times \text{scale factor}$$

Worked Example:

A map has scale $1\text{ cm} : 200\text{ m}$. Two towns are $4.5\text{ cm}$ apart on the map.
$$\text{Real distance} = 4.5 \times 200 = 900\text{ m} = 0.9\text{ km}$$

Finding Map Distance from Real Distance

$$\text{Map distance} = \frac{\text{real distance}}{\text{scale factor}}$$

Worked Example:

Same map ($1\text{ cm} : 200\text{ m}$). A road is $1.4\text{ km}$ long in reality.
$$1.4\text{ km} = 1400\text{ m}$$
$$\text{Map distance} = \frac{1400}{200} = 7\text{ cm}$$

EXAM TIP: Convert units before applying the scale. A scale of \$1:5000$ means $1\text{ cm} = 5000\text{ cm} = 50\text{ m}$.

Compass Directions

Cardinal directions: North (N), South (S), East (E), West (W)

Intercardinal: NE, NW, SE, SW

Compass bearings: Measured clockwise from North, from $0°$ to $360°$.

True bearing: Always written as 3 digits, e.g. $045°$, $135°$, $270°$.

COMMON MISTAKE: Confusing the bearing of $090°$ (East) with going upward. Bearings always start from North and go clockwise.

Reading Floor Plans

Floor plans show the layout of a building from above. Key skills:
- Identify rooms and their dimensions from the scale
- Calculate actual areas using the scale
- Read door/window positions

Example:

A floor plan uses scale \$1:100$. A room measures $3\text{ cm} \times 4\text{ cm}$ on the plan.
$$\text{Actual dimensions} = 3\text{ cm} \times 100 = 300\text{ cm} = 3\text{ m}, \quad 4\text{ cm} \times 100 = 4\text{ m}$$
$$\text{Actual area} = 3 \times 4 = 12\text{ m}^2$$

Reading Street Maps and Topographic Maps

• Use the scale bar to measure distances
• Use a legend/key to interpret symbols
• Use grid references to specify locations (e.g. D4 means column D, row 4)
• Contour lines on topographic maps show elevation — closer contours = steeper slope

VCAA FOCUS: Map and plan questions typically ask you to find a real distance, calculate an area from a plan, identify a direction or bearing, or read a grid reference. Always show the scale conversion step.