Proportional Reasoning with Scale Drawings

Scaling changes every length by the same factor k — which multiplies area by k2 and volume by k3.

A road on a map measures 12 cm. The scale bar says 3 cm = 5 km. How long is the road?

The map is small and the road is not. One number connects them.

a map, a ruler, and a road 3 cm = 5 km the road measures 12 cm the map is small, the road is not — one number connects them

Twelve centimetres is four lots of three. So the road is four lots of five kilometres: 20 km.

Everything on the map shrank by the same factor, which is why one scale works for every road on it.

But be careful with what is inside those shapes. A park 2 cm by 3 cm on the map is not 2 km by 3 km of park.

Lengths scale by k. Areas scale by k2 — because an area is a length times a length, and both of them scaled. For volumes it is k3, for the same reason.

A shape scaled by two: lengths double, area quadruples 3 × 4 · area 12 × 2 6 × 8 · area 48 lengths × 2 area × 4 because 2 × 2 = 2² volume would be × 8

What a scale says

A scale of 1 : 50 means one unit on the drawing stands for 50 of the same units in reality.

A 6 cm line on the drawing is 6 × 50 = 300 cm — that is 3 metres — in real life.

Going the other way, divide: a real length of 12 m is 1200 cm, so on the drawing it measures 1200 ÷ 50 = 24 cm.

Similar figures

Two figures are similar if their angles match and their sides are all in the same ratio.

Find the scale factor from one pair of corresponding sides, then use it on all the others. If one triangle has a side of 6 matching a side of 15, then k = 156 = 2.5 and every other side scales by 2.5.

Match them up first

Before calculating, list the sides in matching order — shortest with shortest, longest with longest. Most errors here come from pairing the wrong sides, not from the arithmetic.

The big idea: area does not scale the same way

Double every length and the area does not double — it quadruples.

A 3 × 4 rectangle has area 12. Scale by 2 to get 6 × 8, area 48. That is 4 times, because both the length and the width grew: 2 × 2 = 4.

Same idea in three dimensions: volume multiplies by k3. Doubling a cube's edge makes it hold 8 times as much.

The k, k2, k3 rule

Scale factor k multiplies:
lengths by k  ·  areas by k2  ·  volumes by k3.
This is true for every shape, not just rectangles.

Indirect measurement

You can measure a tree without climbing it. At the same moment, a 5 ft person casts a 4 ft shadow and the tree casts a 32 ft shadow.

The sun's rays make similar triangles, so 54 = h32, giving h = 40 ft.

Reading maps

A map scale of 1 cm : 2.5 km works the same way. Measure in cm, multiply by 2.5 for km.

Keep the units straight. Convert before you multiply, not after, or the numbers will not mean anything.

Worked examples

Reading a floor plan

On a plan at scale 1 : 40, a room measures 8.5 cm by 6 cm. Find the real dimensions and real area.

  1. Lengths: 8.5 × 40 = 340 cm and 6 × 40 = 240 cm Multiply each length by the scale factor.
  2. In metres: 3.4 m by 2.4 m Divide centimetres by 100.
  3. Real area: 3.4 × 2.4 = 8.16 m2 Compute the area from the real lengths.
  4. Check with the rule: drawing area = 51 cm2, times 402 = 1600, gives 81600 cm2 = 8.16 m2 ✓ Area scales by k2, which confirms the answer.

3.4 m × 2.4 m, area 8.16 m2.

Similar triangles

Triangle ABC has sides 6, 8, 10. Similar triangle DEF has its shortest side = 9. Find the perimeter of DEF.

  1. Scale factor: k = 96 = 1.5 Shortest side to shortest side.
  2. Other sides: 8 × 1.5 = 12 and 10 × 1.5 = 15 Every length scales by the same k.
  3. Perimeter = 9 + 12 + 15 = 36 Or scale the original perimeter: 24 × 1.5 = 36 ✓

36

Scaling area and volume

A model car is 112 the size of the real one. The real car's hood has area 3.6 m2 and the car holds 60 L of air. Find the model's values.

  1. Scale factor k = 112 Every length shrinks by 12.
  2. Area scales by k2 = 1144: 3.6 ÷ 144 = 0.025 m2 Two dimensions shrink at once.
  3. Volume scales by k3 = 11728: 60 ÷ 1728 ≈ 0.035 L Three dimensions shrink at once.

Area 0.025 m2; volume about 0.035 L.

Words to know

Scale factor
The number every length is multiplied by.
Scale drawing
A picture in proportion to the real thing, at a stated scale like 1 : 50.
Similar figures
Same shape, different size: equal angles, proportional sides.
Corresponding sides
Sides in matching positions in two similar figures.

Mistakes to watch for

Multiplying area by k instead of k2 — the classic scale-drawing error. Double every length and you get four times the area, because the length and the width both doubled.

Pairing the wrong corresponding sides. In similar figures the sides must be matched by position, not by which looks longest on the page. Mark the equal angles first, then pair the sides between them.

Mixing units. A scale of 1 cm = 2 m means you must measure in centimetres and answer in metres. Multiplying centimetres by a factor meant for metres is out by a hundred.

Assuming the scale factor runs the direction you want. Going map-to-real multiplies; real-to-map divides. Ask which one should be bigger and check your answer against it.

Every flat map of the round Earth distorts something. The Mercator projection keeps directions true, which is why sailors loved it.

The price is area: on a Mercator map Greenland looks about the size of Africa. Africa is roughly fourteen times larger.

There is no projection that gets shape, area and direction all right at once — it is a theorem, not a lack of effort.

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