Scale Factors and Drawings
A map, a model railway, a phone photo and an architect's plan all show something at the wrong size on purpose. The scale factor says how wrong, and once you know it every length converts the same way. Areas and volumes, though, change by more than you'd expect.
National curriculum KS3: use scale factors, scale diagrams and maps; identify and construct congruent and similar shapes, including enlargement with a given scale factor1 · Enlargement: every length times k
To enlarge a shape by scale factor k, multiply every length by k. The angles don't change, so the shape keeps its shape: the two are similar. Drag k and watch the lengths. A scale factor between 0 and 1 makes it smaller, but it is still called an enlargement.
2 · Area grows by k², volume by k³
Double the sides of a square and you can fit four of the original inside, not two. Area has two dimensions, so it picks up the factor twice. A cube's volume picks it up three times. This is the same squared-and-cubed idea as converting m² to cm², and it trips people up for the same reason.
Why a model boat floats differently. Scale a boat down by 10: lengths ÷ 10, hull area ÷ 100, but volume and weight ÷ 1000. That's why a mouse can fall off a table unharmed and an elephant can't: strength goes with area (k²) while weight goes with volume (k³). Scaling is not neutral.
3 · Reading a scale
A scale like 1 : 25 000 means 1 unit on the map is 25 000 of the same units in real life. Measure in cm on the map, multiply, then convert the answer to something sensible (25 000 cm = 250 m). Scales also appear as "1 cm represents 2 km", which is 1 : 200 000 once both sides are in cm.
| Scale | 1 cm represents | Typical use |
|---|---|---|
| 1 : 100 | 1 m | House floor plan |
| 1 : 1 250 | 12.5 m | Site plan for planning permission |
| 1 : 25 000 | 250 m | OS Explorer walking map (Jersey's official map is 1 : 25 000) |
| 1 : 50 000 | 500 m | OS Landranger |
| 1 : 1 000 000 | 10 km | Road atlas |
| 1 : 76 | 0.76 m | OO gauge model railway |
4 · Make a scale drawing
Going the other way: a real room is too big for paper, so choose a scale that fits. Divide every real length by the scale factor. Pick a room and a scale; the plan redraws, and the drawn lengths are labelled so you can check your own.
5 · Similar shapes: find the missing length
Two shapes are similar if one is an enlargement of the other. Then every pair of matching sides is in the same ratio. To find a missing side: find the scale factor from a pair you know, then apply it. Choose which way round to divide by asking "is the answer bigger or smaller?"
6 · Where scale shows up
Maps and GPS
Pinch-zooming a phone map changes the scale continuously; the little scale bar in the corner redraws so you can still judge distance.
Architecture and engineering
Plans at 1 : 100, details at 1 : 5. A builder reads 35 mm on the plan as 3.5 m on site. Misreading the scale is a famously expensive mistake.
Models and miniatures
Lego minifigures are about 1 : 40; a 1 : 18 model car is 18 times shorter, 324 times less surface, 5832 times less volume than the real one.
Photos and screens
Resizing an image by 50% keeps it similar; stretching only the width doesn't, and faces look wrong. "Maintain aspect ratio" means "use one scale factor".
Biology
Magnification = image size ÷ actual size, the same formula as a scale factor. A ×400 microscope image of a 0.02 mm cell is 8 mm across.
Computing
Vector graphics store shapes, so they scale by any k without blurring; bitmaps store pixels, so enlarging by k means inventing k² − 1 new pixels per original.
Worked examples
A map has scale 1 : 50 000. Two villages are 7 cm apart on it. How far apart are they really?
A photo 15 cm by 10 cm is enlarged so the long side is 45 cm. Find the new short side and the new area.
Draw a 12 m × 9 m garden at a scale of 1 : 150
Exam-style: triangles ABC and PQR are similar. AB = 6 cm, PQ = 9 cm, BC = 8 cm. Find QR.
Quick check
What you've learned
A scale factor multiplies every length; angles stay the same, so the shapes are similar. Area scales by k² and volume by k³. A map scale 1 : n means multiply map lengths by n and then convert units to something sensible; a scale drawing divides real lengths by n. For similar shapes, find the scale factor from a known pair of sides, then multiply or divide, checking whether the answer should come out bigger or smaller.