Learn8300Papers 1–3 · Higher tierGeometry and measuresSimilarity and congruence

Geometry and measures · Subtopic 3 of 17

Similarity and Congruence

Similar shapes are enlargements of each other: same angles, all lengths multiplied by the same scale factor $k$ . Areas then scale by $k^2$ and volumes by $k^3$ . Congruent shapes are identical: $k=1$ .

Cambridge 4024 Paper 1Cambridge 4024 Paper 2Edexcel 4MA1 Papers 1H, 2HEdexcel 4MA1 Papers 1F, 2FAQA 8300 Papers 1–3≈ 30 minRevision 5 · updated 2026-10-27

Section 1 of 14

Learning objectives

Find a length scale factor and use it to find missing sides.
Spot similar triangles, including one inside another.
Use area factor $k^2$ and volume factor $k^3$ .
Know the conditions for congruent triangles: SSS, SAS, ASA (or AAS) and RHS.

Pause and think

A photo is enlarged so that its width doubles. How many copies of the original photo would fit on the enlarged one?

Section 2 of 14

Core idea

Start hereMeasuring a tree with a shadow

A 2 m post casts a 3 m shadow; at the same moment a tree casts a 24 m shadow. The sun's rays make two triangles with the same angles, so the tree is $2\times8=16$ m tall. Same shape, different size: similar.

The big idea

Similar: same angles, sides in ratio $k$ . Areas scale by $k^2$ , volumes by $k^3$ . Congruent: identical (SSS, SAS, ASA, RHS).

1. Similar triangles

Two similar triangles: scale factor 2

Lengths double, but area is multiplied by

2^2=4

: you need four small triangles to cover the big one.

Worked exampleMissing side

Similar triangles: sides 4, 6 and 6, $x$ (matching order).

Goal: $x$ .
Strategy: Scale factor from a matching pair.

$k=\tfrac64=1.5$
Matching sides.
$x=6\times1.5=9$
Scale.

Answer

x=9

✓ Check

4:6=6:9

✓.

2. Area and volume

Why

k^2

and

k^3

$\text{area}=\text{length}\times\text{length}\to(k\ell_1)(k\ell_2)=k^2\ell_1\ell_2$
Two lengths, each scaled by $k$ .
$\text{volume}\to k^3\ell_1\ell_2\ell_3$
Three lengths.

Worked exampleSimilar cylinders

Heights 6 cm and 9 cm. Small surface area 120 cm², volume 160 cm³.

Goal: Large area and volume.
Strategy: $k$ , $k^2$ , $k^3$ .

$k=1.5$
$9\div6$ .
$120\times2.25=270$
Area × $k^2$ .
$160\times3.375=540$
Volume × $k^3$ .

Answer

270

cm²;

540

cm³.

✓ CheckVolume ratio

540/160=3.375=1.5^3

✓.

Worked exampleFrom volume back to length

Similar solids have volumes 27 and 64.

Goal: Length scale factor.
Strategy: Cube root.

$k^3=\tfrac{64}{27}$
Volume ratio.
$k=\tfrac43$
Cube root.

Answer

\frac43

✓ CheckArea ratio would be

\frac{16}9

✓.

Common misconceptionUsing k for area Double the length means 4 times the area, not twice.

3. Congruence

SSS, SAS, ASA (or AAS), RHS.
Not AAA: that only shows similarity.

Pause and think

Map scale 1 : 1000. Real area of 3 cm² on the map?

Summary

Length $k$ , area $k^2$ , volume $k^3$ .
Congruence tests.

Similar: Same shape, different size.
Congruent: Same shape and size.

HistoryMeasuring a pyramid with a shadow. A story told by later Greek writers says Thales measured the height of the Great Pyramid by waiting until his own shadow was as long as he was tall. At that moment, the pyramid's shadow (measured from its centre) equalled its height: similar triangles, with scale factor set by the Sun.

Fun factWhy there are no giant ants. Make an ant

10

times longer and its weight grows

10^3=1000

times, but the cross-section of its legs only grows

10^2=100

times. Its legs would buckle. That square–cube law is why elephants have such thick legs compared with gazelles.

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