Scale Drawings

School may be kilometres away, yet the map fits in your palm; a building stands tens of metres tall, yet its blueprint lies flat on a desk. Drawing the world shrunk by a fixed ratio relies on the skills trained in ratios and proportion — the scale.

A scale is a secret code

1 : 100 reads "one to one hundred" and means: 1 unit on the drawing, 100 units in reality. 1 cm on paper is 100 cm — one metre — in real life; 1 m on paper is 100 m in the world.

The bigger the second number, the harder the shrink: a neighbourhood plan uses 1 : 500, a city map 1 : 100000, and a world map shrinks far harder still.

Reading a plan

On a floor plan at scale 1 : 50, a wall is drawn 2 cm long. How long is the real wall?

$2 \times 50 = 100 \ \mathrm{cm} = 1 \ \mathrm{m}$

A stubby 2 centimetres on paper, a full metre in life — every line on a drawing must be blown back up like this.

Converting both ways: multiply up, divide down

Drawing → real: multiply by the second number. On a 1 : 200 plan you measure 5 cm:

$5 \times 200 = 1000 \ \mathrm{cm} = 10 \ \mathrm{m}$

Real → drawing: divide by the second number. To fit a 12 m wide room onto a 1 : 150 plan:

$12 \ \mathrm{m} = 1200 \ \mathrm{cm}, \qquad 1200 \div 150 = 8 \ \mathrm{cm}$

Match the units before touching the numbers — both ends of a scale must speak the same unit. Mixing metres with centimetres throws the answer off by a factor of a hundred.

Area scales by the square

Length times 100 — so area times what? Not 100, but $100 \times 100 = 10000$ . Check it: a real rug of 2 m × 2 m drawn at 1 : 100 becomes 2 cm × 2 cm. The real area is 4 m², that is 40000 cm²; the drawn area is 4 cm² — exactly 10000 times smaller.

So the area ratio is the square of the length ratio: a 1 : 10 drawing differs 100-fold in area; a 1 : 1000 map differs a million-fold. Length stretches one direction, area stretches two at once — hence the square.

Don't scale area like length

On a 1 : 100 plan a flower bed measures 6 cm². What is the real area? Not 600 cm², but $6 \times 10000 = 60000 \ \mathrm{cm^2}$ — that is, 6 m². Square first, then convert.

Scaling on the coordinate board

InteractiveScale Board

x3y-2

(3, -2) → Quadrant IV

Walk x steps horizontally, then y steps vertically — the ordered pair (x, y) pins down exactly one point.

Treat the coordinate board as the drawing: each point's coordinates are the "centimetres" on paper. To enlarge a figure 3-fold, multiply every coordinate by 3 — same shape, bigger size. This "shape unchanged, size changed" transformation is the home turf of similar figures from congruence and similarity: a 1 : 100 map and the real world are a pair of similar figures differing by 100.

Check yourself

Quick quiz

1. On a 1 : 1000 map two towns sit 4 cm apart. What is the real distance?
2. A room 9 m wide must fit on a 1 : 150 plan. How long is it drawn?
3. A model at scale 1 : 10 — its surface area is what fraction of the real thing's?