Scale drawings

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.

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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.

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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., $005^\circ, 072^\circ, 210^\circ$).

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: $135^\circ$

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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 $180^\circ$.

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: $100^\circ$

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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.

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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.

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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).