Maps, floor plans and models show real things at a different size. A scale such as "1 cm represents 5 m" tells how lengths on the drawing match real lengths. Every scale problem is a proportion.
Key idea
Real length $=$ drawing length $\times$ the scale, and drawing length $=$ real length $\div$ the scale. Areas do not scale the same way: if lengths are multiplied by $k$, areas are multiplied by $k^2$.
Why it works
A rectangle has two lengths, and both of them grow. If each side becomes $k$ times as long, the area becomes $k \times k = k^2$ times as big. On a grid you can see it: a copy at scale 2 holds $2 \times 2 = 4$ copies of the original, and at scale 3 it holds 9.
Worked examples
Worked example: Drawing to real life
A park map uses the scale 1 cm represents 5 m. A path on the map is 4.5 cm long. How long is the real path?
- Each centimeter on the map is 5 m.
- $4.5 \times 5 = 22.5$.
Answer: 22.5 m
Worked example: Real life to drawing
An architect uses 1 cm represents 4 m. How long should an 18 m wall be on the plan?
- Count the groups of 4 m: $18 \div 4 = 4.5$.
- The wall is 4.5 cm long on the plan.
Answer: 4.5 cm
Worked example: Area
On a floor plan with scale 1 cm represents 2 m, a room is 3 cm by 4 cm. What is its real area?
- Real lengths: $3 \times 2 = 6$ m and $4 \times 2 = 8$ m.
- Real area: $6 \times 8 = 48$ square meters.
- Compare: the plan's area is $12$ square cm, and $12 \times 2^2 = 48$ ✓.
Answer: 48 square meters
Common mistake
Multiplying an area by the scale instead of the scale squared. In the last example, $12 \times 2 = 24$ is wrong. Safest habit: change each length to real size first, then find the area.
A scale with no units, like $1 : 50$, means real lengths are 50 times the drawing lengths, in the same units: 1 cm on the drawing is 50 cm in real life.