4.3 BETA

Scale drawings

4 learning objectives • 2 core • 2 extended

1. Overview

Scale drawings are used to accurately represent real-world objects and distances on a smaller scale, such as on maps or architectural plans. They rely on maintaining correct proportions using a scale factor. This topic covers interpreting and creating scale drawings, working with three-figure bearings for direction, and understanding how scales affect lengths and areas. A firm grasp of ratios, unit conversions, and basic geometry is essential for success.

Key Definitions

  • Scale: The ratio that defines the relationship between the distance on a map or drawing and the actual distance in real life.
  • Ratio Scale: A scale written in the form $1 : n$, where 1 unit on the drawing represents $n$ units in real life (both must be the same units).
  • Bearing: A measure of direction expressed as an angle in degrees, measured clockwise from North.
  • Three-figure Bearing: A bearing written using three digits (e.g., $045^\circ$ instead of $45^\circ$).
  • North Line: A vertical line drawn on a map pointing towards the North pole, used as the $000^\circ$ reference point.

Core Content

A. Interpreting and Using Scales

Scales are usually given as a ratio, such as $1 : 50,000$. This means $1\text{ cm}$ on the map represents $50,000\text{ cm}$ in real life.

Conversion Tip: To convert between map units and real-world units, remember:

  • $1\text{ km} = 1,000\text{ m}$
  • $1\text{ m} = 100\text{ cm}$
  • Therefore, $1\text{ km} = 100,000\text{ cm}$

Worked Example 1 — Finding Actual Distance

A map has a scale of $1 : 20,000$. The distance between two towns on the map is $8.5\text{ cm}$. Calculate the actual distance in kilometers.

  1. State the given information: Scale: $1 : 20,000$ Map distance: $8.5\text{ cm}$

  2. Multiply map distance by the scale factor: $8.5 \times 20,000 = 170,000\text{ cm}$ Reason: To find the actual distance in cm.

  3. Convert cm to meters (divide by 100): $170,000 \div 100 = 1,700\text{ m}$ Reason: There are 100 cm in 1 meter.

  4. Convert meters to kilometers (divide by 1,000): $1,700 \div 1,000 = 1.7\text{ km}$ Reason: There are 1,000 meters in 1 kilometer.

Final Answer: $1.7\text{ km}$

Worked Example 2 — Finding Map Distance

The actual distance between two cities is $45\text{ km}$. A map has a scale of $1 : 500,000$. What is the distance between the two cities on the map, in centimeters?

  1. State the given information: Scale: $1 : 500,000$ Actual distance: $45\text{ km}$

  2. Convert the actual distance to centimeters: $45\text{ km} = 45 \times 1,000\text{ m} = 45,000\text{ m}$ Reason: Convert km to meters. $45,000\text{ m} = 45,000 \times 100\text{ cm} = 4,500,000\text{ cm}$ Reason: Convert meters to centimeters.

  3. Set up a proportion: $\frac{1}{500,000} = \frac{\text{Map Distance}}{4,500,000}$ Reason: Express the scale as a fraction.

  4. Solve for the map distance: $\text{Map Distance} = \frac{4,500,000}{500,000} = 9\text{ cm}$ Reason: Multiply both sides by 4,500,000.

Final Answer: $9\text{ cm}$

B. Three-Figure Bearings

Bearings must follow three strict rules:

  1. Measured from North.
  2. Measured Clockwise.
  3. Written with three digits (e.g., $005^\circ, 072^\circ, 210^\circ$).
📊A point A with a vertical North line. An arrow rotates clockwise from the North line to a point B. The angle between North and the line AB is labeled $060^\circ$.

Worked Example 3 — Drawing a Bearing

Draw the position of point $Y$ from point $X$ on a bearing of $120^\circ$ at a distance of $5\text{ cm}$.

  1. Mark point $X$ and draw a vertical North line.
  2. Place the center of the protractor on $X$ with the $0^\circ$ line aligned with the North line.
  3. Measure $120^\circ$ clockwise and mark a point.
  4. Draw a line from $X$ through the mark exactly $5\text{ cm}$ long.
  5. Label the end of the line $Y$.

Worked Example 4 — Measuring a Bearing

Point $B$ is located southeast of point $A$. A North line is drawn at point $A$. Using a protractor, the angle measured clockwise from the North line at $A$ to point $B$ is $135^\circ$. State the three-figure bearing of $B$ from $A$.

  1. Identify the given information: Angle measured clockwise from North at $A$ to $B$: $135^\circ$

  2. Express the bearing as a three-figure bearing: Since the angle is $135^\circ$, the three-figure bearing is $135^\circ$.

Final Answer: $135^\circ$

Extended Content (Extended Only)

A. Area Scale Factors

When a scale is $1 : n$, it refers to lengths. For areas, the scale factor must be squared ($1 : n^2$). This is because area is a two-dimensional measurement, so both the length and width are scaled by the linear scale factor.

Worked Example 5 — Calculating Actual Area

A map has a scale of $1 : 5,000$. A forest on the map has an area of $12\text{ cm}^2$. Calculate the actual area of the forest in square meters ($\text{m}^2$).

  1. State the given information: Scale: $1 : 5,000$ Map area: $12\text{ cm}^2$

  2. Find the linear scale factor in meters: $1\text{ cm} : 5,000\text{ cm}$ $1\text{ cm} : 50\text{ m}$ Reason: Divide 5,000 cm by 100 to convert to meters.

  3. Square the scale factor for area: $(1\text{ cm})^2 : (50\text{ m})^2$ $1\text{ cm}^2 : 2,500\text{ m}^2$ Reason: Area scale factor is the square of the linear scale factor.

  4. Multiply map area by the area scale factor: $12 \times 2,500 = 30,000\text{ m}^2$ Reason: To find the actual area.

Final Answer: $30,000\text{ m}^2$

B. Back Bearings (Reverse Bearings)

To find the bearing of $A$ from $B$ when given the bearing of $B$ from $A$:

  • If the bearing is less than $180^\circ$, add $180^\circ$.
  • If the bearing is more than $180^\circ$, subtract $180^\circ$.
📊Two parallel North lines at points A and B. A line connects them. Show that the interior angles between parallel lines add to $180^\circ$, explaining why the $\pm 180^\circ$ rule works.

Worked Example 6 — Calculating Back Bearing

The bearing of town $B$ from town $A$ is $065^\circ$. Calculate the bearing of town $A$ from town $B$.

  1. State the given information: Bearing of $B$ from $A$: $065^\circ$

  2. Apply the back bearing rule: Since $065^\circ < 180^\circ$, add $180^\circ$. $065^\circ + 180^\circ = 245^\circ$

Final Answer: $245^\circ$

Worked Example 7 — Calculating Back Bearing (Alternative)

The bearing of a ship from a lighthouse is $280^\circ$. Find the bearing of the lighthouse from the ship.

  1. State the given information: Bearing of ship from lighthouse: $280^\circ$

  2. Apply the back bearing rule: Since $280^\circ > 180^\circ$, subtract $180^\circ$. $280^\circ - 180^\circ = 100^\circ$

Final Answer: $100^\circ$

Key Equations

  • Scale Ratio: $1 : n = \frac{\text{Drawing Length}}{\text{Actual Length}}$
  • Area Scale Factor: $(\text{Linear Scale})^2$
  • Volume Scale Factor: $(\text{Linear Scale})^3$
  • Back Bearing: $x \pm 180^\circ$

Formula Sheet Note: These formulas are not provided on the IGCSE formula sheet. You must memorize the conversion factors and the area/volume rules.

Common Mistakes to Avoid

  • ❌ Wrong: Writing a bearing as $45^\circ$. ✓ Right: Always use three digits for bearings: $045^\circ$.
  • ❌ Wrong: Using the length scale factor to convert area (e.g., multiplying $12\text{ cm}^2$ by $5,000$ instead of $5,000^2$). ✓ Right: Always square the scale factor before applying it to an area.
  • ❌ Wrong: Confusing "Bearing of $A$ from $B$" with "Bearing of $B$ from $A$". ✓ Right: Put your pencil on the point following the word "from"—that is where your North line and protractor go.
  • ❌ Wrong: Skipping unit conversions and getting huge, unrealistic numbers. For example, assuming a map distance of 5 cm represents 5 km directly when the scale is 1:1000. ✓ Right: Convert cm to km early in the calculation or use the $100,000$ conversion factor carefully.
  • ❌ Wrong: Rounding intermediate calculations and then using the rounded value in subsequent steps. ✓ Right: Keep exact values throughout the calculation and only round the final answer if necessary.

Exam Tips

  • Command Words:
    • "Measure": Use your ruler or protractor physically on the paper.
    • "Calculate": Use the numbers given; do not rely on your own measurements unless the diagram is "to scale".
  • Calculator vs Non-Calculator: In non-calculator papers, scale factors are often simple multiples (like $200$ or $500$). In calculator papers, expect values like $1 : 25,000$.
  • Accuracy Marks: IGCSE markers allow a small margin of error for measurements (usually $\pm 2\text{ mm}$ for length and $\pm 2^\circ$ for bearings), but you will lose marks if your lines are not sharp or your protractor is misaligned.
  • The "1 : n" Form: If asked to give a scale in the form $1 : n$, ensure the "1" has no units and $n$ is calculated by dividing the real distance (in cm) by the map distance (in cm).

Test Your Knowledge

Ready to check what you've learned? Practice with 10 flashcards covering key definitions and concepts from Scale drawings.

Study Flashcards Practice MCQs

Frequently Asked Questions: Scale drawings

What is Scale in Scale drawings?

Scale: The ratio that defines the relationship between the distance on a map or drawing and the actual distance in real life.

What is Ratio Scale in Scale drawings?

Ratio Scale: A scale written in the form $1 : n$, where 1 unit on the drawing represents $n$ units in real life (both must be the same units).

What is Bearing in Scale drawings?

Bearing: A measure of direction expressed as an angle in degrees, measured clockwise from North.

What is Three-figure Bearing in Scale drawings?

Three-figure Bearing: A bearing written using three digits (e.g., $045^\circ$ instead of $45^\circ$).

What is North Line in Scale drawings?

North Line: A vertical line drawn on a map pointing towards the North pole, used as the $000^\circ$ reference point.

What are common mistakes students make about Scale drawings?

Common mistake: Writing a bearing as $45^\circ$. → Correct: Always use three digits for bearings: $045^\circ$. Common mistake: Using the length scale factor to convert area (e.g., multiplying $12\text{ cm}^2$ by $5,000$ instead of $5,000^2$). → Correct: Always square the scale factor before applying it to an area.