Ratio, Proportion & Rates of Change
Grade 4-5Scale Diagrams & Maps
A map scale is just a ratio between a drawn length and the real length it represents, and once that ratio is set up correctly, converting distances, finding the scale itself, or even scaling an area all follow the same underlying idea. This lesson covers converting between map and real distances, finding the scale of a drawing, and scaling area using the square of the linear scale factor.
Written by Asad, Co-Founder of Teachably
Video walkthrough coming soon
The written lesson below covers everything you need in the meantime.
Understanding scale notation
A scale can be written as a ratio, such as 1:50000, or in the form 1 cm : n km. In both cases, the first number represents a length on the map or diagram, and the second represents the real length it stands for, once units are matched up.
A scale of 1:50000 means every 1 cm on the map represents 50000 cm (which is 500 m) in real life. A scale of 1 cm : 2 km means every 1 cm on the map represents 2 km in real life.
Converting a map distance to a real distance
Multiply the map distance by the scale factor, then convert the answer into a sensible unit, usually km, since scale factors are large.
A map has a scale of 1:50000, and two towns are 8 cm apart on the map. The real distance is 8 × 50000 = 400000 cm, which converts to 4 km.
Converting a real distance to a map distance
Convert the real distance into the same unit as the scale, then divide by the scale factor to find the map distance.
A map has a scale of 1:100000, and the real distance between two towns is 6 km. Converting to cm gives 600000 cm, so the map distance is 600000 ÷ 100000 = 6 cm.
Finding the scale of a drawing
Convert both the drawn length and the real length to the same unit, write them as a ratio, then simplify it into the form 1:n.
A wall of real length 6 m is drawn as a line 3 cm long. Converting 6 m to 600 cm gives the ratio 3:600, which simplifies to 1:200.
Scaling area
While lengths scale by the linear scale factor, area scales by the square of that factor, since area involves two dimensions.
A garden is drawn with a scale of 1:200, and its area on the diagram is 15 cm². The area scale factor is 200² = 40000, so the real area is 15 × 40000 = 600000 cm², which converts to 60 m².
Worked Examples
Three exam-style questions, fully solved.
A map has a scale of 1:50000. The distance between two towns on the map is 8 cm. Find the real distance between the towns in km.
Easy- 1.Multiply the map distance by the scale factor: 8 × 50000
- 2.Convert the result from cm to km: 400000 cm = 4 km
Answer: 4 km
On a scale drawing, a wall of length 6 m is drawn as a line 3 cm long. Find the scale of the drawing in the form 1:n.
Medium- 1.Convert both lengths to the same unit: 6 m = 600 cm
- 2.Write the lengths as a ratio: 3:600
- 3.Simplify the ratio: 1:200
Answer: 1:200
A garden is drawn on a scale diagram with a scale of 1:200. The area of the garden on the diagram is 15 cm². Find the real area of the garden in m².
Hard- 1.Find the area scale factor by squaring the linear scale: 200² = 40000
- 2.Multiply the drawn area by the area scale factor: 15 × 40000 = 600000 cm²
- 3.Convert the result from cm² to m²: 600000 cm² = 60 m²
Answer: 60 m²
Avoid These
The most common mistakes students make.
Forgetting to convert units, such as cm to km, before or after applying the scale factor.
Not fully simplifying a scale written in the form 1:n, or mixing up units within the ratio.
Multiplying by the scale factor when converting a real distance to a map distance, instead of dividing.
When scaling an area, using the linear scale factor directly instead of squaring it first.
In multi-step problems, such as those involving speed and time, forgetting an earlier conversion step before using the distance in a later calculation.
FAQ
Questions parents and students ask.
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