BTT Mathematics / Primary Mathematics Learning Hub / Scale Drawings and Maps
A scale drawing represents a real object with every corresponding length changed by the same factor. A map is a scale drawing of locations and routes. Enlargement and reduction use the same mathematics: corresponding lengths are multiplied by one scale factor.
If one centimetre on a plan represents five metres in reality, a seven-centimetre wall on the plan represents thirty-five metres. The calculation works because the same length relationship is used everywhere on the drawing.
This guide is Primary Mathematics enrichment. It connects ratio, measurement, coordinates and geometry. It is not a claim that every enlargement, scale-factor or area-scaling idea below is required at every Primary level. Use the MOE Primary curriculum page and the learner’s current school scope as the official reference point.
Meaning of scale · Maps and real distance · Enlargement and reduction · Perimeter and area scaling · Coordinates and dimensions · 24 questions · Worked answers · Teaching and transfer
1. A scale compares corresponding lengths
A scale of 1:50 means one unit on the drawing corresponds to fifty of the same units in reality. If the drawing is measured in centimetres, 1 cm represents 50 cm. The ratio does not mean one centimetre automatically represents fifty metres.
Worked example A: Convert before comparing
A model wall is 8 cm long and represents a real wall 2 m long. Convert 2 m to 200 cm. The ratio is 8:200=1:25. The scale is 1:25.
Worked example B: Drawing to reality
A rectangular plan measures 6 cm by 4 cm at scale 1:50. Real dimensions are 300 cm by 200 cm, or 3 m by 2 m.
Worked example C: Reality to drawing
At scale 1 cm:400 m, a real route of 2.8 km equals 2,800 m. The map length is 2,800÷400=7 cm.
A scale is not an area ratio
At scale 1:50, lengths are divided by fifty. Areas are divided by 50²=2,500. This is why a floor-plan rectangle cannot have its area converted by multiplying by only fifty.
2. Map distances require units and route interpretation
A straight-line map distance represents the straight-line distance between points under the map model. A road route may bend, so its measured map length can be longer than the straight-line separation.
Worked example D: Ratio scale 1:50,000
A route measures 4 cm. Real length is 4×50,000=200,000 cm=2,000 m=2 km.
Worked example E: Scale 1 cm:250 m
A path measures 3.6 cm. Real distance is 3.6×250=900 m.
Worked example F: Find map length
A real distance is 3 km and the map scale is 1 cm:500 m. Three kilometres is 3,000 m, so map length is 6 cm.
Always state whether a measurement is a map length or real length. The same numeral can refer to very different physical quantities.
3. Enlargement and reduction multiply every corresponding length by one factor
An enlargement factor of three changes every length to three times its original. A reduction factor of one-half changes every length to half its original.
Worked example G: Enlargement
A segment 5 cm long is enlarged by factor 3. The new length is 15 cm.
Worked example H: Reduction
A length 18 cm is reduced by factor 0.5. The new length is 9 cm.
Worked example I: Find the scale factor
A 12 cm side becomes 8 cm. Scale factor=8÷12=2/3.
Worked example J: Preserve shape, not size
A 10 cm by 6 cm photograph is reduced so its width becomes 7.5 cm. The scale factor is 7.5÷10=0.75. Height becomes 6×0.75=4.5 cm.
If one dimension is changed by a different factor from another, the shape has been stretched rather than scaled uniformly.
4. Perimeter follows the length factor; area follows its square
If every length is multiplied by k, every boundary segment is multiplied by k, so perimeter is multiplied by k. Area measures two-dimensional coverage, so it is multiplied by k².
Worked example K: Rectangle enlarged by factor two
A 4 cm by 7 cm rectangle has perimeter 22 cm and area 28 cm². Enlarging by factor two gives dimensions 8 cm by 14 cm, perimeter 44 cm and area 112 cm². Perimeter doubled; area multiplied by four.
Worked example L: Triangle area under factor three
A triangle has area 18 cm². After enlargement factor three, its area is 18×9=162 cm².
Worked example M: Reduction factor one-third
A square has area 81 cm². Reducing every length by one-third multiplies area by one-ninth, giving 9 cm².
Worked example N: Recover the length factor from area
An area changes from 10 cm² to 160 cm². Area factor=16, so positive length factor is 4.
5. Coordinates can record a scale transformation
When an enlargement is centred at the origin, multiplying both coordinates by the same factor moves each point along the ray from the origin while preserving shape.
Worked example O
Point (2,3) enlarged by factor two about the origin becomes (4,6).
Worked example P
Point (−3,4) enlarged by factor three about the origin becomes (−9,12).
This coordinate rule depends on the stated centre being the origin. A different centre requires measuring from that centre first.
Scale drawings of rooms
A room measures 5.4 cm by 3.6 cm on a 1:100 plan. Real dimensions are 5.4 m by 3.6 m because each centimetre represents one hundred centimetres, or one metre. Real area is 19.44 m².
6. Practice: 24 original questions
Questions 1–8: Scale drawings and maps
1. A plan uses 1 cm:5 m. A wall measures 7 cm on the plan. Find the real length.
2. A map uses 1 cm:250 m. A path measures 3.6 cm. Find the real distance.
3. A map scale is 1:50,000. A route measures 4 cm. Find the real distance in kilometres.
4. A real road is 3 km long. On a map, 1 cm represents 500 m. Find its map length.
5. A model length is 8 cm and the real length is 2 m. Express the scale as 1:n.
6. A rectangle measures 6 cm by 4 cm on a 1:50 drawing. Find its real dimensions in metres.
7. A map scale is 1:25,000. A route measures 12 cm. Find the real distance in kilometres.
8. A real route is 2.8 km. A map uses 1 cm:400 m. Find the map length.
Questions 9–16: Enlargement, reduction and scale factor
9. A 5 cm segment is enlarged by factor 3. Find the new length.
10. An 18 cm segment is reduced by factor 0.5. Find the new length.
11. A 4 cm by 7 cm rectangle is enlarged by factor 2. Find its new perimeter and area.
12. A triangle has area 18 cm². It is enlarged by factor 3. Find the new area.
13. A square has area 81 cm². It is reduced by length factor 1/3. Find the new area.
14. An area changes from 10 cm² to 160 cm² under a uniform enlargement. Find the positive length scale factor.
15. A shape has perimeter 30 cm. It is enlarged by factor 1.5. Find the new perimeter.
16. A side length changes from 12 cm to 8 cm under a uniform reduction. Find the scale factor.
Questions 17–24: Coordinates, area and checking
17. Enlarge point (2,3) by factor 2 about the origin.
18. Enlarge point (−3,4) by factor 3 about the origin.
19. A 10 cm by 6 cm photograph is uniformly reduced to width 7.5 cm. Find the new height.
20. A room is 5.4 cm by 3.6 cm on a 1:100 plan. Find its real dimensions and area.
21. A road measures 6.5 cm on a 1:20,000 map. Find the real road length in kilometres.
22. A model car is 12 cm long and represents a 4.8 m car. Express the scale as 1:n.
23. A drawing at scale 1:50 has area 24 cm². Find the corresponding real area in square metres.
24. A learner says: “Doubling every side length doubles the area.” Is the statement correct? Explain.
7. Worked answers
Answers 1–8
1. 35 m. 7×5=35.
2. 900 m. 3.6×250=900.
3. 2 km. 4×50,000=200,000 cm=2 km.
4. 6 cm. 3,000 m÷500 m per cm=6 cm.
5. 1:25. 2 m=200 cm; 8:200=1:25.
6. 3 m by 2 m. 6×50=300 cm and 4×50=200 cm.
7. 3 km. 12×25,000=300,000 cm=3 km.
8. 7 cm. 2.8 km=2,800 m; divide by 400.
Answers 9–16
9. 15 cm. 5×3.
10. 9 cm. 18×0.5.
11. Perimeter 44 cm; area 112 cm². New dimensions 8 by 14.
12. 162 cm². Area factor is 3²=9.
13. 9 cm². Area factor is (1/3)²=1/9.
14. 4. Area factor is sixteen, whose positive square root is four.
15. 45 cm. Perimeter follows the length factor: 30×1.5.
16. 2/3. 8÷12=2/3.
Answers 17–24
17. (4,6). Multiply both coordinates by two.
18. (−9,12). Multiply both coordinates by three.
19. 4.5 cm. Scale factor=7.5÷10=0.75; 6×0.75=4.5.
20. 5.4 m by 3.6 m; 19.44 m². At 1:100, one centimetre represents one metre.
21. 1.3 km. 6.5×20,000=130,000 cm=1.3 km.
22. 1:40. 4.8 m=480 cm; 12:480=1:40.
23. 6 m². Area factor is 50²=2,500. Real area=24×2,500=60,000 cm²=6 m².
24. No. Doubling every length multiplies area by 2²=4. The statement confuses a one-dimensional factor with a two-dimensional measure.
8. Teaching and transfer
If a learner multiplies a 1:50 map measurement by fifty metres instead of fifty centimetres, ask them to write the scale with explicit units first. Ratio scales use the same unit on both sides before conversion.
When area is scaled linearly
Draw a 1×1 square and enlarge it by factor two. The new square is 2×2 and contains four unit squares. This makes the squared area factor visible.
When one dimension is changed incorrectly
Compare a uniform reduction of a 10×6 photograph with changing only its width. The second result changes proportions and is not a scale copy.
When coordinates become mechanical
Plot the original and enlarged point together with the origin. The enlarged point should lie on the same ray from the centre. This gives a geometric check beyond multiplying coordinates.
Connect maps to rate and units
A map scale behaves like a constant conversion rate between drawing length and real length. The Unitary Method guide can support learners who need to find one centimetre’s real value before scaling.
Continue through this enrichment collection
For changing patterns over stages, use Recursive Sequences, Growing Patterns and Figurate Numbers. For route structures, use Networks, Routes, Nodes and Traversal Puzzles. For classification diagrams, use Sets, Venn Diagrams and Classification.
Return to the BTT Primary Mathematics Learning Hub.
Original enrichment guide with 24 original practice questions and separate worked answers. Map and model scales are mathematical examples, not surveying instructions or official map data.