Secondary 3 · Mathematics · Space
Similar figures and scale
Recognise similar shapes, find scale factors, and use them to calculate unknown lengths, including in scale drawings and maps.
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These lessons were written around Australian Curriculum strands and matched to Singapore levels by age (Primary 1 is the first year of primary school). They are not aligned to MOE syllabuses or the PSLE, so check them against the syllabus your child is working towards.
Learning goals
Learners will
- Recognise similar figures as enlargements or reductions with equal angles and proportional sides.
- Find and use scale factors to calculate unknown lengths in similar figures.
- Use scale ratios to interpret scale drawings and maps.
Success looks like
- I can explain why two figures are or are not similar.
- I can find a scale factor and use it to calculate a missing side.
- I can convert a measurement on a map or plan to a real distance using a scale such as 1 : 50 000.
The big idea
Two figures are similar if one is an enlargement (or reduction) of the other. In similar figures, matching angles are equal and matching sides are in the same ratio. A photo and its enlargement are similar; a square and a long thin rectangle are not.
The scale factor tells you how many times bigger the image is: scale factor = image length ÷ original length. A scale factor greater than 1 is an enlargement; between 0 and 1 is a reduction. If a triangle with sides 3, 4 and 5 is enlarged to 6, 8 and 10, the scale factor is 2.
To check whether two figures are similar, divide each pair of matching sides. If every ratio is the same, the sides are in proportion. For triangles, it is also enough to show that all the angles match.
To find a missing length:
- Match up the corresponding sides (it helps to redraw the shapes facing the same way).
- Find the scale factor from a pair of sides you know.
- Multiply (to go to the bigger figure) or divide (to go to the smaller figure).
Scale drawings and maps use a scale ratio. A map scale of 1 : 50 000 means 1 cm on the map represents 50 000 cm in real life. Multiply the map distance by 50 000, then convert units: 100 cm = 1 m and 100 000 cm = 1 km. A floor plan at 1 : 100 is convenient because 1 cm on the plan is 100 cm = 1 m in real life.
Similar triangles let us measure things we can't reach. At the same time of day, a pole and a tree make similar triangles with their shadows, so we can find the tree's height from the length of its shadow.
Worked examples
Finding a missing side
Two rectangles are similar. The smaller is 4 cm by 6 cm. The larger has a shorter side of 10 cm. Find its longer side.
- The shorter sides correspond: scale factor = 10 ÷ 4 = 2.5.
- Longer side = 6 × 2.5 = 15 cm.
- Check: 4 : 6 and 10 : 15 both simplify to 2 : 3.
Answer: 15 cm
Using a map scale
On a map with scale 1 : 50 000, two towns are 6 cm apart. How far apart are they in real life?
- Real distance = 6 × 50 000 = 300 000 cm.
- 300 000 cm ÷ 100 = 3000 m.
- 3000 m ÷ 1000 = 3 km.
Answer: 3 km
Shadows and similar triangles
A 1.5 m pole casts a 2 m shadow. At the same time, a tree casts a 12 m shadow. How tall is the tree?
- The sun's rays make the same angle, so the triangles are similar.
- Scale factor (shadows) = 12 ÷ 2 = 6.
- Tree height = 1.5 × 6 = 9 m.
Answer: 9 m
Practice check
Have a go, then check your answers. Each answer comes with an explanation.
Answer guide for parents
Which of these pairs are ALWAYS similar?
Any two squares — All squares have 90° angles and four equal sides, so one is always an enlargement of the other. Rectangles, triangles and parallelograms can have different shapes.
A side of 5 cm is enlarged to 15 cm. What is the scale factor?
3 — Scale factor = image ÷ original = 15 ÷ 5 = 3.
On a map with scale 1 : 25 000, what real distance does 4 cm represent?
1 km — 4 × 25 000 = 100 000 cm = 1000 m = 1 km.
In similar figures, matching angles are:
equal — An enlargement changes lengths but not angles, so matching angles are equal.
Two triangles are similar. The smaller has sides 4 cm, 6 cm and 8 cm. The shortest side of the larger triangle is 10 cm. How long is the longest side of the larger triangle?
20 cm — Scale factor = 10 ÷ 4 = 2.5. Longest side = 8 × 2.5 = 20 cm.
A 1.2 m stick casts a 1.6 m shadow. At the same time, a building casts a 20 m shadow. How tall is the building, in metres?
15 m — Scale factor = 20 ÷ 1.6 = 12.5. Height = 1.2 × 12.5 = 15 m.
A floor plan has a scale of 1 : 200. A room is 3.5 cm long on the plan. How long is the real room, in metres?
7 m — 3.5 × 200 = 700 cm = 7 m.
A model car is built to a scale of 1 : 18. The real car is 4.5 m long. How long is the model, in centimetres?
25 cm — 4.5 m = 450 cm. Model length = 450 ÷ 18 = 25 cm.
Watch out for
- Adding instead of multiplying: thinking a 4 × 6 rectangle enlarged so 4 becomes 10 makes 6 become 12 (adding 6). Similar figures use a scale factor, which multiplies.
- Matching the wrong sides. Redraw the figures in the same orientation and pair the shortest with the shortest, longest with longest.
- Forgetting to convert units with map scales, such as giving 300 000 km instead of 3 km. Write the units at every step.
Tips for parents
- Use a map on a family outing: measure a distance on the map and convert it with the scale, then compare with the real trip distance.
- Photocopy or scan a picture at 200% and measure to show that lengths double but angles stay the same.
- Do the shadow activity outdoors on a sunny day — it is a memorable way to measure a tree or building.
Go further
- Draw a scale floor plan of your home or a room at 1 : 50 or 1 : 100, including furniture.
- Investigate what happens to area under enlargement: if lengths are multiplied by 2, what happens to the area? (It is multiplied by 4.)
- Build a simple model of a building or vehicle to a chosen scale, recording your scale calculations.
About 45 minutes
You'll need
- Ruler, protractor and grid paper
- A road map or printed map with a scale
- A tape measure (for the shadow activity)
Key words
- Similar figures
- Figures with the same shape but not necessarily the same size: matching angles are equal and matching sides are in the same ratio.
- Enlargement
- A transformation that makes a figure bigger or smaller by multiplying every length by the same number.
- Scale factor
- The number every length is multiplied by to go from the original figure to the image: image length ÷ original length.
- Corresponding sides
- Sides in matching positions in two similar figures.
- Scale (ratio)
- The ratio of a length on a drawing or map to the real length, such as 1 : 100.
Still tricky?
For parents
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