Understand the idea.
Lengths share one factor
A length scale factor compares matching lengths in a fixed direction. A plan means each drawing length is multiplied by to obtain the real length, provided both use the same unit.
Convert units before forming a scale ratio. The scale factor itself has no unit because matching length units cancel. State whether you are moving from drawing to reality or from the original object to its enlargement.
Use correspondence, not appearance
Similar objects preserve shape because every corresponding length uses the same positive factor. Match width to width and height to height; the labelled dimensions, rather than the apparent size of a sketch, determine the calculation.
An enlargement without distortion therefore multiplies both width and height equally. Stretching only one direction changes the shape. Also distinguish lengths from areas: an area multiplies two lengths and therefore uses the square of the length factor.
See the method in action.
Room plan
A 1:50 plan shows a room length of 8 cm. Find real length in metres.
Worked solution
- 1
Apply the drawing-to-real factor.
A scale gives fifty real centimetres for each plan centimetre.
- 2
Convert to the requested unit.
There are centimetres in a metre.
- 3
Check the scale in common units.
The drawing and real lengths reproduce the stated ratio.
4 m
Photograph enlargement
A 12 cm wide, 8 cm high picture is enlarged to width 18 cm without distortion. Find its new height.
Worked solution
- 1
Find the width enlargement factor.
New width divided by original width gives the original-to-new factor.
- 2
Apply the same factor to height.
An undistorted enlargement preserves all corresponding length ratios.
- 3
Check that proportions agree.
Width and height have been scaled equally.
12 cm
Try it yourself.
Check your understanding
A 1:100 plan shows a length of 3.5 cm. Find the real length.
Show answer & reasoning
3.5 m
The real length is . The factor is applied before converting centimetres to metres.
Keep practising
6 worksheet questions · 13 marks
Read the questions online
- A plan shows a 6 cm wall. Find real length.2 marks
- A model boat is 32 cm long. Find real length.2 marks
- Enlarge a 15 cm length by factor 1.2. Find the new length.2 marks
- A 9 m wall is drawn as 18 cm. Find the scale.2 marks
- Similar triangles have matching sides 4 and 10 cm. A 7 cm side of the smaller matches what larger length?2 marks
- A 20 by 12 cm picture is reduced to width 15 cm without distortion. Find height and justify your factor.3 marks
Read the worked solutions
Equivalent correct methods earn the corresponding marks. Include units where asked.
- A plan shows a 6 cm wall. Find real length.2 marks
cm m.
12 m - A model boat is 32 cm long. Find real length.2 marks
cm m.
8 m - Enlarge a 15 cm length by factor 1.2. Find the new length.2 marks
.
18 cm - A 9 m wall is drawn as 18 cm. Find the scale.2 marks
900 cm compared with 18 cm gives .
- Similar triangles have matching sides 4 and 10 cm. A 7 cm side of the smaller matches what larger length?2 marks
.
.
17.5 cm - A 20 by 12 cm picture is reduced to width 15 cm without distortion. Find height and justify your factor.3 marks
for all corresponding lengths.
.
9 cm
Curriculum reference & lesson scope
Australian Curriculum v9 and SA Curriculum
Surface area of composite solids; volume of composite solids; proportion/scaling of objects; related length/area/volume factors; measurement error; the effect of rounding and repeated approximations; practical design models; evaluating assumptions and suitable precision.
Official curriculum source ↗