Practice Maths

Scale Drawings and Maps

Key Ideas

Key Terms

scale
Written as map distance : actual distance (e.g. 1 : 50 000 means 1 unit on the map equals 50 000 of the same unit in reality).
same unit
Before calculating.
Finding actual distance
Actual = map distance × scale denominator.
Finding map distance
Map distance = actual distance ÷ scale denominator.
Floor plans and engineering drawings
Work the same way but at much smaller scales (e.g. 1 : 100 means 1 cm on plan = 1 m in real life).
Hot Tip Always write the scale with the same unit on both sides before using it. For example, 1 : 50 000 means 1 cm : 50 000 cm = 1 cm : 500 m = 1 cm : 0.5 km. Choose whichever form makes the calculation easiest.

Worked Example

Question: A map has a scale of 1 : 25 000. The distance between two towns on the map is 3.6 cm. Find the actual distance in metres.

Step 1 — Identify the scale:   1 cm on map = 25 000 cm in reality.

Step 2 — Calculate actual distance in cm:   3.6 × 25 000 = 90 000 cm

Step 3 — Convert to metres:   90 000 cm ÷ 100 = 900 m

What is a Scale Drawing?

A scale drawing is an accurate drawing where every length has been reduced (or enlarged) by the same ratio. Maps, architectural plans, and engineering diagrams are all scale drawings. The scale tells you the relationship between the drawing and reality. For example, a scale of 1:50 000 means that 1 unit on the map equals 50 000 of those same units in real life. So 1 cm on the map = 50 000 cm = 500 m = 0.5 km in reality.

Scales are written as ratios (e.g. 1:200 000), as fractions (e.g. 1/200 000), or as statements (e.g. "1 cm represents 2 km"). These all mean the same thing.

Converting Map Distances to Real Distances

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

Formula: Real distance = Map distance × Scale factor

Example: A map has a scale of 1:25 000. Two towns are 6 cm apart on the map. What is the real distance?

Real distance = 6 cm × 25 000 = 150 000 cm = 1500 m = 1.5 km

Always convert to a sensible unit at the end — for large distances, kilometres are more appropriate than centimetres.

Converting Real Distances to Map Distances

To find how far something should be drawn on a map, divide the real distance by the scale factor.

Formula: Map distance = Real distance ÷ Scale factor

Example: A road is 3 km long. Draw it on a map with scale 1:50 000. How long should the line be?

Convert 3 km to cm first: 3 km = 300 000 cm. Then: Map distance = 300 000 ÷ 50 000 = 6 cm.

The key step is making sure both distances are in the same unit before dividing.

Scale and Area

When a scale affects lengths, it affects areas by the scale factor squared. If the length scale is 1:k, then the area scale is 1:k2.

Example: A paddock measures 4 cm × 3 cm on a map with scale 1:10 000. What is its real area?

Real dimensions: 4 × 10 000 = 40 000 cm = 400 m, and 3 × 10 000 = 30 000 cm = 300 m.

Real area = 400 m × 300 m = 120 000 m2 = 12 ha.

Alternatively: Map area = 4 × 3 = 12 cm2. Scale factor for area = 10 0002 = 100 000 000. Real area = 12 × 100 000 000 cm2 = 1 200 000 000 cm2 = 120 000 m2. Same answer.

Designing Your Own Scale Drawing

When creating a scale drawing (e.g. a floor plan of your bedroom), follow these steps: (1) Measure all real dimensions. (2) Choose a convenient scale that fits the drawing on your page — for a 5 m room on A4 paper, a scale of 1:50 works well (5 m = 500 cm ÷ 50 = 10 cm on paper). (3) Convert every real measurement to a drawing measurement. (4) Draw accurately with a ruler and label your scale.

A good scale drawing must include the scale written clearly (e.g. "Scale: 1:50") and a title.

Key tip: The most common mistake is forgetting to convert units before multiplying or dividing. Always get both measurements into the same unit (usually centimetres) before applying the scale. Write out the conversion step explicitly — it takes 5 seconds and prevents errors worth marks.

Mastery Practice

  1. Use the given scale to find the actual distance. Show your unit conversions clearly. Fluency

    1. Scale 1 : 10 000. Map distance: 4.5 cm. Find the actual distance in metres.
    2. Scale 1 : 50 000. Map distance: 7.2 cm. Find the actual distance in kilometres.
    3. Scale 1 : 200. Map distance: 3 cm. Find the actual distance in metres.
    4. Scale 1 : 100 000. Map distance: 12.5 cm. Find the actual distance in kilometres.
    5. Scale 1 : 25 000. Map distance: 8 mm. Find the actual distance in metres.

  2. Find the map distance (in cm) for each actual distance using the given scale. Fluency

    1. Scale 1 : 50 000. Actual distance: 3 km.
    2. Scale 1 : 200. Actual distance: 14 m.
    3. Scale 1 : 100 000. Actual distance: 4.5 km.
    4. Scale 1 : 10 000. Actual distance: 750 m.
    5. Scale 1 : 25 000. Actual distance: 2 km.

  3. Find the scale of each drawing, expressed as a ratio 1 : n. Fluency

    1. A 2 cm drawing represents 500 m.
    2. A 5 cm drawing represents 1 km.
    3. A 4 mm drawing represents 2 m.
    4. An 8 cm drawing represents 40 m.
    5. A 3 cm drawing represents 1.5 km.

  4. A house floor plan has a scale of 1 : 100. Understanding

    1. On the plan, the lounge room is 5.5 cm long and 4 cm wide. Find the actual dimensions.
    2. The real kitchen is 3.6 m × 2.8 m. What are the plan dimensions in centimetres?
    3. Find the actual area of the lounge room. (Use the real dimensions.)
    4. Find the plan area of the lounge room (in cm²). How many times smaller is the plan area than the real area?
    5. The total floor area of the house is 18 m × 12 m. What is the plan area of the whole house in cm²?

  5. A topographic map has a scale of 1 : 25 000. Understanding

    1. Two towns are 14.8 cm apart on the map. Find the actual distance in kilometres.
    2. A lake appears as a rectangle 3.2 cm × 1.4 cm on the map. Find the actual area of the lake in km².
    3. A road between two points is 22.5 km long. How many centimetres does it measure on the map?
    4. A student says “1 cm² on this map represents 625 m² of real land.” Is this correct? Verify by calculation.

  6. An engineering drawing of a machine component uses the scale 2 : 1 (the drawing is larger than the real part). Understanding

    1. A bolt appears 4.8 cm long on the drawing. What is its actual length?
    2. A hole has a diameter of 1.5 cm on the drawing. What is the actual diameter?
    3. A surface on the drawing measures 6 cm × 3 cm. Find the actual area of the surface.
    4. Why would engineers use a scale larger than 1 : 1 for small components?

  7. A city planner is designing a new suburb. On the planning map (scale 1 : 2000), a rectangular park is drawn as 7.5 cm × 4 cm. Problem Solving

    1. Find the actual dimensions of the park.
    2. Find the actual area of the park in m² and in hectares (1 ha = 10 000 m²).
    3. A path runs diagonally across the park. On the map it measures 8.6 cm. Find the actual length of the path.
    4. Grass turf costs $12 per m². Find the cost of turfing the entire park.
    5. A fence is built around the entire perimeter of the park. Fencing costs $85 per metre. Find the total fencing cost.

  8. A hiker is planning a cross-country route using a map with scale 1 : 50 000. The route consists of three legs. Problem Solving

    Leg 1: 6.4 cm on map (heading north)  |  Leg 2: 9.2 cm on map (heading east)  |  Leg 3: 11.3 cm on map (returning diagonally)

    1. Find the actual length of each leg in kilometres.
    2. Find the total distance of the hike in kilometres.
    3. The hiker walks at an average pace of 4 km/h. How long (in hours and minutes) will the hike take?
    4. Leg 3 should be the hypotenuse of a right triangle formed by legs 1 and 2. Use Pythagoras’ theorem to verify whether the map distances are consistent (show working).

  9. An architect builds a scale model of a house at 1 : 40. Problem Solving

    1. The model is 36 cm long and 22 cm wide. Find the actual length and width of the house.
    2. The model roof is 18 cm tall. Find the actual roof height.
    3. A window in the model is 1.5 cm × 2 cm. Find the actual window dimensions.
    4. Find the actual floor area of the house.
    5. Carpet costs $45 per m². Find the cost to carpet the entire floor area (assuming a single rectangular level).

  10. Two maps show the same region. Map A has scale 1 : 10 000 and Map B has scale 1 : 50 000. Problem Solving

    1. A road is 3.5 km long. How long does it appear on Map A (in cm)?
    2. How long does the same road appear on Map B (in cm)?
    3. Which map shows more detail? Explain your reasoning.
    4. A forest appears as 48 cm² on Map A. Find the actual area of the forest in km².
    5. How large would the same forest appear on Map B (in cm²)?
See Answers ➔