Similarity is not “the shapes look alike”. It is a precise statement that corresponding angles match and corresponding lengths are in a constant ratio. Congruence is stronger: corresponding lengths are equal, so the scale factor is exactly 1.
This twenty-third Secondary 4 Mathematics Learning Guide develops congruence, similarity, scale drawings and area-volume scaling as one geometric system. It belongs to the Secondary Mathematics Hub and S1–S4 Capability Map.
Current syllabus connection: upper-secondary Geometry and Measurement includes congruence, similarity, scale drawings, corresponding length relationships and the way area and volume change under enlargement. The examples below are original teaching material.
Congruent figures preserve size as well as shape
Two congruent figures have the same shape and the same size. One may be translated, rotated or reflected, but corresponding lengths and corresponding angles remain equal.
Congruence is therefore a rigid-fit idea. If one figure can be moved without stretching so that it lies exactly on the other, the figures are congruent.
Similarity preserves shape while allowing scale
Two similar figures have equal corresponding angles and proportional corresponding lengths. If every length in the second figure is twice the matching length in the first, the linear scale factor is 2.
Similarity preserves angle. Scale factor changes length.
Correspondence must be established before ratios are written
Suppose triangle ABC is similar to triangle PQR. If A corresponds to P, B to Q and C to R, then AB corresponds to PQ, BC to QR and AC to PR.
A correct ratio uses matching sides in the same order. Mixing a long side from one figure with a non-corresponding side from the other can produce a plausible-looking but meaningless scale factor.
Worked Example 1 | Find a missing corresponding side
Two similar triangles have corresponding sides 6 cm and 9 cm. Another side of the smaller triangle is 10 cm. Find the corresponding side of the larger triangle.
Scale factor from smaller to larger:
9/6 = 3/2.
So the corresponding larger side is:
10×3/2 = 15 cm.
Length ratios become squared area ratios
If every length is multiplied by k, an area uses two length dimensions, so the area is multiplied by k².
If the length scale factor is 3, the area scale factor is 3²=9.
Worked Example 2 | Area ratio from length ratio
Two similar figures have corresponding length ratio 4:7. Find their area ratio.
Area ratio = 4²:7² = 16:49.
Do not use 4:7 for area. The figure has been scaled in two perpendicular dimensions.
Volume ratios cube the length factor
Volume uses three dimensions. If every length is multiplied by k, the volume is multiplied by k³.
Worked Example 3 | Volume ratio from length ratio
Two similar solids have corresponding length ratio 2:5. Find their volume ratio.
Volume ratio = 2³:5³ = 8:125.
This is why a modest change in length can create a much larger change in capacity.
Recover the length factor by taking roots
If an area ratio is known, take square roots to recover the corresponding length ratio. If a volume ratio is known, take cube roots.
Worked Example 4 | Recover scale from area
Two similar shapes have area ratio 81:121. Find their length ratio.
Length ratio = √81:√121 = 9:11.
Worked Example 5 | Recover scale from volume
Two similar containers have volume ratio 27:64. Find their corresponding length ratio.
Length ratio = ∛27:∛64 = 3:4.
Scale drawings translate between drawing length and real length
A scale such as 1:50,000 means 1 unit on the drawing represents 50,000 of the same units in reality. Units must be made consistent before the final answer is interpreted.
At scale 1:50,000, 1 cm on a map represents 50,000 cm=500 m=0.5 km in reality.
Worked Example 6 | Map distance
A map uses scale 1:25,000. Two points are 7.2 cm apart on the map. Find the actual distance in kilometres.
Actual distance = 7.2×25,000=180,000 cm.
Since 100,000 cm=1 km:
180,000 cm = 1.8 km.
Scale drawings can also work backwards
If an actual road is 3.6 km long and the map scale is 1:60,000, convert 3.6 km to centimetres first: 3.6 km=360,000 cm. Then drawing length=360,000/60,000=6 cm.
Similar triangles often appear inside one larger diagram
Parallel lines frequently create equal corresponding or alternate angles, which can establish similarity. Once similarity is proved, corresponding sides may be used proportionally.
The proof step should come before the ratio step. A ratio is only valid after correspondence has been justified.
Worked Example 7 | Parallel lines create similar triangles
In triangle ABC, D lies on AB and E lies on AC, with DE parallel to BC. Suppose AD=6 cm, AB=10 cm and AE=7.5 cm. Find AC.
Because DE is parallel to BC, triangles ADE and ABC are similar. Their corresponding length ratio is:
AD/AB = 6/10 = 3/5.
Therefore AE/AC=3/5:
7.5/AC=3/5.
AC = 7.5×5/3 = 12.5 cm.
Congruence can prove equal lengths or angles
When two triangles are shown congruent from sufficient matching information, corresponding parts are equal. This can be used to justify a length or angle that was not directly given.
The exact school notation for congruence criteria may vary by context, but the reasoning should identify enough matching side-angle information to establish that only one triangle size and shape is possible.
Worked Example 8 | Congruence through rigid information
Triangles ABC and DEF have AB=DE=5 cm, BC=EF=7 cm and AC=DF=9 cm. The three corresponding sides are equal, so the triangles are congruent. Therefore corresponding angles such as ∠ABC and ∠DEF are equal.
No scale factor other than 1 is involved.
Area and volume can reveal hidden scale factors
If two similar solids have volumes 432 cm³ and 1458 cm³, the volume ratio simplifies to 8:27. Taking cube roots gives length ratio 2:3.
The corresponding surface-area ratio is then 4:9. One comparison can therefore determine several others once the correct dimensional power is used.
Worked Example 9 | Find a missing surface area from volume information
Two similar solids have volume ratio 64:125. The smaller has surface area 96 cm². Find the larger surface area.
Volume ratio 64:125 gives length ratio 4:5. Therefore area ratio is 16:25.
Let larger area be A:
96/A = 16/25, so A=96×25/16=150 cm².
Similarity does not guarantee equal area
Two figures can have exactly the same angles but very different areas. Similarity preserves proportional shape, not size. Congruence is the special case where the length scale factor is 1, which makes area and volume scale factors 1 as well.
Composite situations: scale only what belongs to the similar figures
A larger diagram may contain a pair of similar triangles plus unrelated lengths or regions. Do not apply the similarity scale factor automatically to every measurement on the page.
First identify the two similar figures and their corresponding parts. Then apply the ratio only within that relationship.
Common failure modes
| Error | Cause | Repair |
|---|---|---|
| Uses non-corresponding sides in a proportion | Shape orientation distracts from matching | Mark corresponding angles first |
| Uses length ratio for area | Dimensional change ignored | Square the length factor |
| Uses square of length ratio for volume | Three dimensions compressed to two | Cube the length factor |
| Converts map scale after multiplying inconsistently | Units mixed | Keep one unit through the ratio, convert at end |
| Assumes figures are similar because they look similar | Visual appearance treated as proof | Establish angle and side correspondence |
| Calls similar figures congruent | Scale factor not checked | Congruence requires scale factor 1 |
Verification strategies
- Write correspondence explicitly before forming ratios.
- Check whether all calculated corresponding length ratios agree.
- Square a length ratio for area and cube it for volume.
- Check map units before interpreting the final distance.
- Use triangle inequality when a missing side has been found.
- Ask whether the larger figure actually received a scale factor greater than 1.
Independent practice
- Two similar figures have length ratio 3:8. Find their area ratio.
- Two similar solids have length ratio 5:7. Find their volume ratio.
- Two similar shapes have area ratio 25:81. Find their length ratio.
- Two similar solids have volume ratio 216:343. Find their length ratio.
- A map scale is 1:40,000. A route measures 9.5 cm. Find the actual distance in kilometres.
- Two similar triangles have corresponding sides 8 cm and 14 cm. Another side of the smaller triangle is 12 cm. Find the matching larger side.
- Two similar solids have volume ratio 8:27. The smaller surface area is 72 cm². Find the larger surface area.
Explained answers
1. Area ratio=9:64.
2. Volume ratio=125:343.
3. Length ratio=√25:√81=5:9.
4. Length ratio=∛216:∛343=6:7.
5. 9.5×40,000=380,000 cm=3.8 km.
6. Scale factor=14/8=7/4. Larger side=12×7/4=21 cm.
7. Volume ratio 8:27 gives length ratio 2:3, hence area ratio 4:9. Larger area=72×9/4=162 cm².
Teaching sequence: correspondence before scale
Begin with figures in different orientations and ask learners to match corresponding vertices without using measurements. Then establish length scale factors and only afterwards move into area and volume powers.
Add scale drawings once unit conversion is stable. Finish with composite diagrams and parallel-line similarity, where the learner must justify similarity before calculating.
Connect this guide to Geometry, Trigonometry and Measurement as a Constraint System and Mensuration, Radians and Composite Solids.
Final thought
Similarity is a rule about what survives when scale changes. Angles stay fixed. Lengths scale once, areas twice and volumes three times. Congruence is the no-scaling case.
Match the parts first. Then let dimension decide how the scale factor grows.
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