MISSION · Geometry
Similar Triangles
Same shape, different size, and what that lets you measure
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The idea in one sentence
Two triangles are similar when their angles match, and matching angles force every pair of corresponding sides into the same ratio.
Similar against congruent
Congruent means identical — same angles, same lengths. You could cut one out and lay it exactly over the other.
Similar means same shape, any size. A photo and its enlargement are similar: every angle survives, every length scales by the same factor.
Congruent triangles are similar ones with a scale factor of 1.
Angles are enough
For triangles specifically, matching two angles is enough to prove similarity. The third comes free, since all three must add to \(180°\) — so if two match, the third can’t help but match too. This is the AA test, and it’s the one you’ll use most.
There’s no such shortcut for other shapes. A square and a rectangle have four right angles each and are not similar. Triangles are rigid in a way other polygons aren’t, which is why they hold up bridges and why their angles determine their shape completely.
The scale factor
Divide any length in the big triangle by the matching length in the small one. That number is the scale factor, and it’s the same whichever pair you pick — which is precisely what lets you find a missing side.
If the sides 6 and 9 correspond, the factor is 1.5, so every other length in the larger triangle is 1.5 times its partner. Set it up as a proportion and solve:
\[\frac{x}{4} = \frac{9}{6} \quad \Rightarrow \quad x = 6\]The one thing to get right is which sides correspond. Match them by the angles they sit opposite, not by their position on the page — a rotated or flipped triangle is still similar, and lining up the drawings by eye is how corresponding sides get mismatched.
What it’s for
This is the mathematics of measuring things you can’t reach.
A stick 1 m tall casts a 1.5 m shadow at the same moment a tree casts one 18 m long. The sun’s rays arrive at the same angle for both, so the two triangles are similar:
\[\frac{h}{18} = \frac{1}{1.5} \quad \Rightarrow \quad h = 12 \text{ m}\]The same reasoning gives you the heights of pyramids, the distance to a ship from shore, and the scale on every map you’ve ever used. It is also where trigonometry comes from: the sine of an angle is a fixed number precisely because all right triangles with that angle are similar.
Areas scale differently
Double every length and the area goes up four times, not two — area involves two dimensions, so it scales by the factor squared. Volume scales by the factor cubed.
\[\text{lengths} \times k \quad \Rightarrow \quad \text{areas} \times k^2 \quad \Rightarrow \quad \text{volumes} \times k^3\]Common mix-ups
- Mismatching corresponding sides. Pair them by opposite angles.
- Adding instead of multiplying. Similar shapes scale by a factor, they don’t grow by a constant.
- Scaling area by the length factor. Square it.
- Assuming equal angles make any two shapes similar. True for triangles, not in general.
Check yourself
Two similar triangles have corresponding sides of 4 and 10. If the smaller has another side of 6, what is its partner? And how do their areas compare?
The factor is 2.5, so the partner is 15. The areas differ by \(2.5^2 = 6.25\) times.
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