A scale drawing is a mathematical promise: every represented length changes by the same factor. Once that promise is understood, distances, perimeters and floor-plan dimensions become proportional. Areas require one extra step because two independent lengths have been scaled.
This Secondary 3 Mathematics Learning Guide develops map scales, floor plans, scale drawings, length and area factors, reverse scale calculations, unit conversion and composite-plan reasoning. It extends the earlier ratio-and-proportion guide by concentrating on the decisions that arise when a representation is not the same size as the object it describes.
Official scope: the 2027 SEC G3 Mathematics syllabus K310 includes map scales for distance and area. Its real-world context notes also include floor plans and navigation. This guide uses original teaching examples; real maps and professional drawings may include conventions beyond the simplified mathematical models used here.
Scale routine: make the units compatible → identify drawing-to-actual direction → apply the length factor → square that factor for areas → convert final units → check whether the answer should be larger or smaller than the drawing.
Useful prior guide: Ratio, Proportion, Percentage, Rate and Speed in Real Contexts.
What 1 : n Means
A scale of 1:50 means one unit on the drawing represents fifty of the same units in reality. If 1 cm on a floor plan represents 50 cm in the room, then 4.2 cm represents 210 cm, or 2.10 m.
The ratio itself is unitless only after both quantities are expressed in the same unit. Writing 1 cm : 50 m as 1:50 would be wrong. Since 50 m=5000 cm, that relationship is actually 1:5000.
Worked Example 1: Plan Length to Actual Length
A floor plan is drawn at scale 1:80. A wall measures 6.5 cm on the plan. Find its actual length.
Actual length=6.5×80=520 cm=5.20 m.
The actual wall should be longer than the drawing because the denominator 80 indicates a reduction. A result of 0.08125 cm would reveal that the scale direction had been reversed.
Worked Example 2: Actual Length to Plan Length
A classroom is 9.6 m long. It is represented at scale 1:120. Find the plan length in centimetres.
Convert 9.6 m to 960 cm. Plan length=960/120=8 cm.
Dividing before unit conversion can produce a number with unclear meaning. Put both lengths in the same unit before using a numerical scale ratio.
A Statement Scale Can Be Converted to a Ratio Scale
Suppose a map states “1 cm represents 2.5 km”. Convert 2.5 km to centimetres: 2.5 km=2500 m=250,000 cm.
The ratio scale is therefore 1:250,000. This conversion makes later area-scale reasoning possible because the linear factor is now explicit.
Worked Example 3: Map Distance
A map uses scale 1:50,000. Two points are 7.4 cm apart on the map. Find their represented straight-line ground distance.
Ground distance=7.4×50,000=370,000 cm=3700 m=3.7 km.
This is the distance represented by the measured map segment. A real walking or driving route can be longer if it does not follow that straight line. Do not silently replace one interpretation with the other.
Length Factor and Area Factor Are Not the Same
If every length is multiplied by k, every area is multiplied by k² because area contains two length dimensions.
At scale 1:100, actual lengths are 100 times plan lengths. Actual areas are 100²=10,000 times plan areas, once corresponding units are used consistently.
This is the most important non-linear idea in map scale. A rectangle that becomes 100 times longer and 100 times wider does not become 100 times larger in area; it becomes 10,000 times larger.
Worked Example 4: Plan Area to Actual Area
A room occupies 24 cm² on a plan drawn at scale 1:50. Find the actual floor area in square metres.
Area factor=50²=2500. Actual area=24×2500=60,000 cm².
Since 1 m²=10,000 cm², actual area=6 m².
A common error is to divide 60,000 by 100 because 100 cm=1 m. Area units require the conversion factor to be squared: 1 m²=(100 cm)²=10,000 cm².
Worked Example 5: Actual Area to Plan Area
A hall has actual area 96 m² and is represented at scale 1:200. Find its plan area in cm².
Convert 96 m² to 960,000 cm². Area factor from plan to actual is 200²=40,000.
Plan area=960,000/40,000=24 cm².
The plan area is much smaller, which is consistent with a reduction scale.
Recover the Linear Scale From an Area Scale
If corresponding areas are in ratio 1:3600, the corresponding positive length ratio is 1:√3600=1:60.
Taking the square root is necessary because area recorded two applications of the same linear scale factor.
Worked Example 6: Find the Map Scale From Area Information
A rectangular region has actual area 32.4 km² and area 81 cm² on a map. Find the ratio scale.
Convert actual area to cm². Since 1 km=100,000 cm, 1 km²=10¹⁰ cm². Hence 32.4 km²=3.24×10¹¹ cm².
Area ratio map:actual = 81 : 3.24×10¹¹ = 1 : 4×10⁹.
Linear scale denominator=√(4×10⁹)=20,000√10≈63,245.6. Therefore the approximate ratio scale is 1:63,246 to the nearest whole number.
The awkward result is mathematically valid because the supplied areas were not chosen to create a neat integer scale. Do not force the denominator to a familiar map scale without permission.
Floor Plans Need Every Wall Interpreted Before Areas Are Added
A floor plan may represent an L-shaped room, several adjacent rooms or internal voids. Before calculating area, decide which plan regions belong to the requested space.
The scale factor applies to every linear dimension, but the geometry still determines which lengths are needed and which areas should be added or subtracted.
Worked Example 7: Composite Floor Plan
An L-shaped room is drawn at scale 1:100. On the plan it can be seen as a 7 cm by 5 cm rectangle with a 2 cm by 2 cm corner removed. Find the actual floor area.
Plan area=7×5−2×2=35−4=31 cm².
Area factor=100²=10,000. Actual area=310,000 cm²=31 m².
An equally valid route converts the plan lengths first: 7 m by 5 m minus 2 m by 2 m. Both give 35−4=31 m². Two-route agreement is a strong check.
Worked Example 8: Recover a Missing Plan Dimension
A rectangular room is shown at scale 1:75. Its actual area is 27 m². The plan width is 6 cm. Find the plan length.
Plan width 6 cm represents 450 cm=4.5 m actual width. Actual length=27/4.5=6 m.
Convert 6 m to 600 cm, then plan length=600/75=8 cm.
A second route converts actual area into plan area: 27 m²=270,000 cm²; divide by 75²=5625 to get 48 cm². Then plan length=48/6=8 cm.
Perimeter Uses the Linear Scale, Not the Area Scale
Perimeter is a sum of lengths. If linear scale factor is k, perimeter scales by k. Area scales by k².
This distinction matters in questions combining floor area with skirting, fencing or boundary length. Do not square the factor merely because the shape is two-dimensional; square it only when the measured quantity itself is area.
Worked Example 9: Area and Boundary in the Same Plan
A rectangular garden measures 4 cm by 2.5 cm on a 1:250 plan. Find its actual area and perimeter.
Actual dimensions: 4×250=1000 cm=10 m; 2.5×250=625 cm=6.25 m.
Area=10×6.25=62.5 m². Perimeter=2(10+6.25)=32.5 m.
The answer contains different dimensions: square metres for area and metres for perimeter. Units help keep the scale powers separated.
Scale Bar and Ratio Scale Can Cross-Check Each Other
A printed map may contain a scale bar. If printing or screen resizing changes the physical size of the page, a written ratio scale can become unreliable unless the map is reproduced at the intended size, while a scale bar printed with the map changes proportionally with the image.
For school questions, use the scale information stated by the problem. In practical reading, check whether the document has been resized before treating a measured centimetre as the original printed centimetre.
Worked Example 10: Compare Two Drawings of the Same Object
Drawing A uses scale 1:50 and Drawing B uses scale 1:100 for the same room. A wall is 8 cm on Drawing B. How long is it on Drawing A?
Drawing B represents actual length 8×100=800 cm. At scale 1:50, plan length=800/50=16 cm.
Drawing A has twice the linear size because its reduction is only half as strong. Its corresponding area would be four times the area on Drawing B.
Common Errors and Their First Repair
Mixing units inside the scale ratio: convert both lengths to the same unit before simplifying.
Multiplying when the task requires drawing size: identify whether you are moving from plan to actual or actual to plan.
Using linear factor for area: square the scale factor because two lengths are scaled.
Converting cm² to m² by dividing by 100: use 10,000 because 1 m²=10,000 cm².
Using straight-line map distance as road distance: report exactly what the measured segment represents unless route information is supplied.
Scaling area before identifying the correct composite region: solve the geometry first, then apply the scale consistently.
Independent Practice
1. A plan uses scale 1:60. A wall is 7.5 cm on the plan. Find actual length in metres.
2. An actual corridor is 18 m long and is drawn at 1:150. Find the plan length in centimetres.
3. Convert “1 cm represents 3 km” to a ratio scale.
4. A 1:25,000 map shows two points 9.2 cm apart. Find represented straight-line ground distance in kilometres.
5. A shape has plan area 18 cm² at scale 1:40. Find actual area in m².
6. An actual area is 45 m² at scale 1:100. Find plan area in cm².
7. Corresponding areas are in ratio 1:6400. Find the positive linear scale ratio.
8. An L-shaped plan at 1:100 is a 6 cm by 5 cm rectangle with a 2 cm by 1.5 cm corner removed. Find actual area.
9. A rectangular plan at 1:80 has width 5 cm and actual area 25.6 m². Find plan length.
10. A garden measures 3.2 cm by 2.4 cm on a 1:250 plan. Find actual area and perimeter.
11. The same room appears on drawings at 1:40 and 1:100. A wall is 6 cm on the 1:100 drawing. Find its length on the 1:40 drawing.
12. Explain why a 1:100 floor-plan area of 30 cm² does not represent 3000 cm² of actual area.
Explained Answers
1. 7.5×60=450 cm=4.5 m.
2. 18 m=1800 cm; 1800/150=12 cm.
3. 3 km=300,000 cm, so 1:300,000.
4. 9.2×25,000=230,000 cm=2.3 km, so 2.3 km.
5. Area factor=1600. 18×1600=28,800 cm²=2.88 m².
6. 45 m²=450,000 cm²; divide by 10,000 to get 45 cm² plan area.
7. √6400=80, so 1:80.
8. Plan area=30−3=27 cm². At 1:100, actual area=27 m².
9. Plan width 5 cm represents 4 m actual. Actual length=25.6/4=6.4 m=640 cm. Plan length=640/80=8 cm.
10. Actual dimensions are 8 m and 6 m. Area=48 m²; perimeter=28 m.
11. Actual wall=600 cm. At 1:40, plan length=15 cm.
12. Area uses the squared factor: 100²=10,000. The actual area is 30×10,000=300,000 cm²=30 m².
How to Know the Scale Reasoning Has Transferred
Change the representation instead of only changing the numbers. Move from a ratio scale to a statement scale, from a simple rectangle to an L-shaped floor plan, or from finding a length to recovering a scale from area information. A learner who understands the scale system will decide whether the factor should be used once, squared or reversed.
Parents and teachers can ask three diagnostic questions: What does one centimetre represent? Are we finding a length or an area? Are we moving from drawing to reality or reality to drawing? Those questions expose the mathematical decision without performing the calculation for the student.
Continue the Secondary 3 Learning Route
Continue with Fractional Equations, Restrictions and Equation Recovery, Financial Mathematics: Taxation, Instalments, Bills and Currency Exchange, and Data Collection, Classification, Tabulation and Representation Choice.
A scale answer is secure when the units, direction, dimensional power and geometry all refer to the same representation. Return to the Secondary Mathematics Hub.
