Lesson Guide
Explanation
A scale drawing shrinks reality by a fixed ratio. Scale 1 : 100 means 1 unit on the drawing equals 100 of the SAME unit in real life — so 1 cm on a house plan is 100 cm (a metre) of actual wall. A 4 cm wall on the plan is a 4 m wall in the house. The ratio has no units, which is exactly its power: it works in cm, mm or inches alike. Typical scales: house plans 1:100, street maps 1:10000, country maps 1:1,000,000.
Your Turn
On a 1 : 50 plan, a table is 3 cm long. Real length? Check: 150 cm = 1.5 m.
Watch Out
The same unit on BOTH sides: 1 : 100 is not "1 cm = 100 m" — that would be a scale of 1 : 10000.
Explanation
Map to reality: MULTIPLY by the scale. A 6.4 cm route on a 1 : 25000 map is 6.4 × 25000 = 160,000 cm in the world — which nobody says aloud, so convert: ÷100 → 1600 m → 1.6 km. The unit conversion isn't an optional extra; it IS the second half of every scale question. Useful anchor to memorise: on a 1 : 100,000 map, 1 cm = 1 km exactly.
Your Turn
A 1 : 50000 map shows a trail of 9 cm. Real length in km? Check: 450,000 cm = 4.5 km.
Watch Out
cm to km is ÷100,000 (two hops: ÷100 then ÷1000). One missing zero turns a stroll into a marathon.
Explanation
Drawing reality onto a plan runs the machine backwards: DIVIDE by the scale. A 6 m wall on a 1 : 200 plan: convert to plan units first (600 cm), then divide — 600 ÷ 200 = 3 cm on paper. The direction check saves you every time: heading to the big world, multiply; heading to the small paper, divide. And sanity-check the size: plan distances are small — if your answer wouldn't fit on a sheet of paper, you went the wrong way.
Your Turn
A 15 km highway on a 1 : 500,000 map — how many cm? Check: 1,500,000 cm ÷ 500,000 = 3 cm.
Watch Out
Two separate jobs: unit conversion AND scale division. Doing them in one mental step is where zeros vanish.
Explanation
Third variant: the scale itself is missing. A 12 m boat drawn 6 cm long — what's the scale? Same units first (12 m = 1200 cm), write drawing : real (6 : 1200), simplify to 1 : n → 1 : 200. Drawing side first, reduced to 1 — that's the convention. Once found, the scale unlocks every other measurement on the same drawing: the boat's 2 m cabin will be 1 cm on paper.
Your Turn
A 9 km river appears 3 cm on a map. The scale? Check: 3 : 900,000 → 1 : 300,000.
Watch Out
Drawing : real, in that order. Writing 200 : 1 describes an ENLARGEMENT — used for insect diagrams, not house plans!
Explanation
A scale factor k multiplies every LENGTH by k — but areas grow by k2. Triple a rectangle's sides (2 × 1.5 → 6 × 4.5) and the area jumps from 3 to 27: nine times, because two directions each stretched by 3. Consequences everywhere: a 1 : 100 plan shrinks areas by 10000; a pizza twice the diameter is four times the pizza. When a question mentions AREA on a scale drawing, the k2 alarm must ring.
Your Turn
A photo is enlarged with scale factor 4. Its area grows by what factor? Check: 16.
Watch Out
The most-failed question in this topic: "sides doubled, area doubles" — NO, area quadruples. Lengths ×k, areas ×k2.
Explanation
The complete drill: plan→real multiplies, real→plan divides, and a missing scale comes from drawing : real simplified to 1 : n — with unit conversion woven through all three, and the k2 rule waiting whenever area appears. Routine for every question: match the units, pick the direction, calculate, sense-check the size (real things are big, plans are small). Quiz and exercises await!
Your Turn
Workout: 1:20000 map, 7.5 cm track — real km? And a 30 m tower drawn at 1:500 — plan cm? Check: 1.5 km; and 6 cm.
Watch Out
The scale direction never changes — 1 : n always reads drawing-to-real. Only YOUR direction of travel changes.
📌 Key Points — this lesson in a nutshell
- A scale of 1 : n means 1 unit on the plan is n of the SAME units in real life.
- Plan → real: multiply by n. Real → plan: divide by n.
- To find a scale: write both lengths in the same units, then simplify to 1 : n.
- Lengths scale by k, but areas scale by k2.
- Unit conversions are half the battle: 1 m = 100 cm, 1 km = 100,000 cm.
Step 1: Introduction
Step 1 of 6
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