Scales, plans and models: QCE Essential Mathematics Unit 3
“Interpret and use scales written as ratios and scale factors, calculate actual and scaled lengths, read house plans and maps, and create simple scale drawings and models”
A scale 1:n means 1 unit on the drawing is n units in real life. Multiply drawing lengths by n to get real lengths and divide real lengths by n to get drawing lengths, then convert units. Read plans, elevations and maps this way, calculate real areas from real lengths, and choose a clear, convenient scale when drawing.
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What this dot point is asking
Builders, designers and map readers all work from scale drawings. You need to understand what a scale means, convert between drawing (scaled) lengths and real lengths, read plans and maps, and create your own scale drawing.
The answer
Understanding scales
A scale compares a length on a drawing with the real length.
- Ratio form: 1:100 means 1 unit on the drawing represents 100 of the same units in real life. The second number is the scale factor.
- Statement form: "1 cm represents 5 m" can be converted to a ratio by writing both parts in the same unit: 1 cm to 500 cm, so 1:500.
- Drawing to real: multiply by .
- Real to drawing: divide by .
- Convert to sensible units at the end (cm to m, cm to km).
A scale smaller than 1:1 (such as 1:100) makes things smaller (plans, maps). A scale such as 10:1 enlarges (a diagram of a tiny part).
Plans and elevations
- Floor plans show a building from above: walls, doors (arcs show the swing), windows, room names and often dimensions in millimetres.
- Elevations show a building from the side (north, south, east, west).
- Site plans show the building on its block of land.
Real areas come from real lengths: convert each length first, then multiply. (Area scales by , so converting the drawing's area directly is a common error.)
Maps
Map scales are often large, such as 1:25 000 or 1:100 000, or shown as a scale bar. Measure the distance on the map, multiply by the scale factor, and convert to kilometres. Remember straight-line distance on a map is usually shorter than the road distance.
Creating a scale drawing
- Measure or find the real dimensions.
- Choose a scale that fits your page and is easy to work with.
- Divide each real length by the scale factor.
- Draw accurately with a ruler, and label the scale and key dimensions.
On a map with a scale of 1:50 000, a walking track measures 7.4 cm. Walkers average 4 km per hour. How long will the walk take?
- Real distance = 7.4 × 50 000 = 370 000 cm.
- Convert: 370 000 cm = 3700 m = 3.7 km.
- Time = 3.7 divided by 4 = 0.925 hours = 0.925 × 60 = about 56 minutes.
- Note: tracks wind and climb, so allow extra time.
- Dividing when you should multiply
- Real objects are bigger than their plan (for scales like 1:100).
- Forgetting to convert units
- 450 cm is 4.5 m, not 450 m.
- Scaling area with the scale factor
- Convert lengths first, then calculate the area.
Practice questions
Original practice questions graded from foundation to exam level, each with a full worked solution. Try them before revealing the solution.
foundation3 marksA floor plan has a scale of 1:100. A bedroom measures 3.6 cm by 3.2 cm on the plan. Find the real dimensions and floor area of the bedroom.Show worked solution →
Real length = 3.6 × 100 = 360 cm = 3.6 m.
Real width = 3.2 × 100 = 320 cm = 3.2 m.
Area = 3.6 × 3.2 = 11.52 m².
Marking guide: 1 mark for each real dimension, 1 mark for the area.
core4 marksA model car is built at a scale of 1:24. The model is 18.5 cm long. (a) How long is the real car in metres? (b) The real car is 1.8 m wide. How wide should the model be, to the nearest millimetre?Show worked solution →
(a) Real length = 18.5 × 24 = 444 cm = 4.44 m.
(b) 1.8 m = 180 cm. Model width = 180 divided by 24 = 7.5 cm = 75 mm.
Marking guide: 2 marks for (a) (multiply by the scale factor, convert), 2 marks for (b) (convert, divide by the scale factor).
exam5 marksTom is designing a shed that will be 4.8 m by 3 m. He wants to draw it on an A4 page with a drawing area of 25 cm by 18 cm, using a scale of the form 1:n where n is a multiple of 10. Choose the most suitable scale, give the drawing dimensions, and justify your choice.Show worked solution →
Real sizes: 480 cm by 300 cm.
Try 1:10: 48 cm by 30 cm, too big for 25 cm by 18 cm.
Try 1:20: 24 cm by 15 cm, which fits (24 is less than 25 and 15 is less than 18).
Try 1:30: 16 cm by 10 cm, fits but is smaller and less detailed.
Choose 1:20, giving a drawing of 24 cm by 15 cm. It is the largest scale of this form that fits the page, so the drawing is as clear and detailed as possible, and 1:20 makes conversions easy (each 1 cm on the drawing is 20 cm in real life).
Marking guide: 1 mark for converting to cm, 1 mark for testing scales, 1 mark for choosing 1:20, 1 mark for the drawing dimensions, 1 mark for justification.