Scale drawings & similarity
The idea
A map of a city and the city itself share a shape but not a size — that is the heart of scale and similarity. A scale like 1 cm : 25 km is a promise that every map centimeter stands for 25 real kilometers, no exceptions. Two figures are similar when one is a uniform enlargement or reduction of the other: all matching angles equal, all matching lengths multiplied by the same scale factor. Your skills with multiplication, division, and ratios do all the computation.
The working rule is multiplicative both ways: map to real multiplies by the scale factor, real to map divides by it. The trap is additive thinking — reasoning that since one side grew by 10, every side grows by 10. Scaling multiplies; a 2 × 3 rectangle scaled by factor 3 becomes 6 × 9, not 12 × 13. One more subtlety worth storing: lengths scale by the factor k, but areas scale by k², because area stretches in two directions at once.
Worked example
A hiking map uses the scale 1 cm : 25 km. A trail on the map measures 6.4 cm. How long is the real trail, and how many centimeters on this map would represent a 90 km highway?
- The scale promises every map centimeter stands for 25 real kilometers, so multiply to go from map to reality: 6.4 × 25 = 160 km of real trail.
- Quickly estimate to keep yourself honest: 6 cm would be 150 km and 0.4 cm would be 10 km, so 160 km fits perfectly.
- Going from reality to the map reverses the operation, so divide: 90 ÷ 25 = 3.6 cm of map distance for the highway.
- Verify the round trip: 3.6 cm × 25 km per cm = 90 km, which recovers the original highway length, confirming both directions of the conversion.
Answer. The trail is 160 km long in reality, and the 90 km highway would span 3.6 cm on the map.
Check your understanding
- Why does scaling multiply every length by the same factor instead of adding the same amount to each?
- How do you decide whether a conversion calls for multiplying or dividing by the scale factor?
- If a drawing is enlarged by a factor of 3, why does its area grow by a factor of 9 rather than 3?
- What measurements would you compare to test whether two triangles are really similar?
Build the foundations first
Scale drawings & similarity builds on these concepts. If any feel shaky, start there.