MISSION · Geometry
Scale Drawings and Maps
Shrinking the world onto a page without distorting it
LEVEL 1 · EXPLORE
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LEVEL 2 · LEARN
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The idea in one sentence
A scale is a ratio telling you how many real units one drawn unit stands for, applied identically to every length in the picture.
Reading a scale
\(1 : 50\) means one unit on the drawing is fifty of the same unit in reality. A 1 cm line represents 50 cm; a 6 cm line represents 300 cm, which is 3 m.
The units must match on both sides. That’s what makes the ratio unitless and lets it work however you measure — a scale of \(1:50\) is the same instruction whether you’re holding a metric or an imperial ruler.
Maps often use a bigger number. \(1 : 25{,}000\) means 1 cm on the map is 25,000 cm on the ground — that’s 250 m, since 100 cm make a metre. Converting the awkward number into sensible units is usually the first thing to do.
Going each way
- Drawing to real: multiply by the scale.
- Real to drawing: divide by the scale.
A 4 cm wall at \(1:50\) is \(4 \times 50 = 200\) cm, or 2 m. A real 3 m door at the same scale is \(300 \div 50 = 6\) cm on the paper.
Check the direction against common sense every time. The real thing is bigger, so the real number should be bigger. If your answer says a house is 8 cm across, you divided when you should have multiplied.
Every length, and only lengths
A scale drawing is similar to the real thing: all lengths scaled by the same factor, all angles unchanged. That’s what stops the picture distorting. Halve the width but not the height and you’ve drawn something that isn’t the same shape.
But areas do not scale by the same number. At \(1:50\), a room drawn 2 cm by 3 cm — 6 cm² on the page — is really 1 m by 1.5 m, which is 1.5 m², or 15,000 cm². That’s \(50^2 = 2500\) times bigger, not 50.
The rule, as with any similar figures: lengths by \(k\), areas by \(k^2\), volumes by \(k^3\). Forgetting the square is the most expensive mistake in this topic, because it’s the one that turns up in real estimates of paint, flooring, and turf.
Where it’s used
Floor plans, maps, model kits, engineering drawings, dress patterns. Also microscope images and diagrams of atoms, where the scale runs the other way — the drawing is bigger than the object, and the scale is written like \(500:1\).
Common mix-ups
- Multiplying when you should divide. Real is bigger.
- Mismatched units. Convert to a single unit before applying the scale.
- Scaling area by \(k\) instead of \(k^2\).
- Assuming a bigger scale number means a bigger drawing. \(1:100\) is smaller on paper than \(1:50\), because more reality is squeezed into each centimetre.
Check yourself
On a \(1:200\) plan, a garden is drawn 7 cm long. How long is it really? And what would a real 5 m fence measure on the plan?
\(7 \times 200 = 1400\) cm, which is 14 m. The fence is \(500 \div 200 = 2.5\) cm on paper.
LEVEL 3 · REFLECT
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