Divide a side of the second triangle by the matching side of the first. That single number scales every length. Areas scale by its SQUARE — a factor of 2 makes the triangle four times the area, not twice — and perimeters scale by the factor itself.
What similar means
Two triangles are similar when they have the same shape and possibly different sizes: all three angles equal, and all three pairs of sides in the same ratio.
Those two conditions are not independent. For triangles, equal angles force proportional sides and proportional sides force equal angles — which is why AA is enough to establish similarity and why you never need to check all six things.
This is special to triangles. Two quadrilaterals can have all four angles equal and completely different shapes: a square and a non-square rectangle both have four right angles. Triangles are rigid in a way other polygons are not.
Why areas square
Scale a triangle by 2 and the area goes up by 4. Scale by 3 and it goes up by 9. Every time.
The reason is that area involves two lengths multiplied together. Area = ½ × base × height, and if both base and height double, the product goes up by 2 × 2.
So a 2× enlargement uses 4× the paint, 4× the tiles and 4× the fabric. This is where scaling estimates go wrong, and the error is always in the same direction: underestimating.
Perimeters, being lengths, scale linearly — 2× the fencing for a 2× enlargement. It is worth keeping the two straight, because a job usually needs both numbers.
The three similarity tests
You never have to check everything. Any one of these establishes similarity.
AA — two equal angles. The third follows from the 180° sum, so two is enough. This is the one used most, because angles are often easier to establish than lengths — parallel lines give equal angles for free.
SSS — three sides in the same ratio. No angle needed. This is what the page checks when you give more than one pair.
SAS — two sides in ratio with the included angle equal. The angle has to be the one BETWEEN the two sides, exactly as in triangle congruence.
There is no “SSA” similarity test, for the same reason SSA does not fix a triangle: two sides in ratio and a non-included angle can describe two different shapes.
Similar or congruent
Congruent means identical: same shape AND same size. Similar means same shape, any size.
So congruent is the special case of similar where the scale factor is exactly 1. Every congruent pair is similar; almost no similar pair is congruent.
The tests mirror each other. SSS congruence needs three EQUAL sides; SSS similarity needs three sides in the same RATIO. AAA is not a congruence test at all — equal angles fix the shape and say nothing about the size.
Where it is used
Shadows. A stick of known height and its shadow form a triangle similar to the one made by a building and its shadow. This is how Thales is said to have measured the Great Pyramid, and it still works.
Maps and drawings. A scale drawing is a similar figure. Every length on the page relates to reality by one factor — and every AREA by its square, which is why a 1:100 plan represents 10,000 times the floor area.
Photography and optics. Object, lens and image form similar triangles, which is where the magnification formula comes from.
Trigonometry itself. Sine, cosine and tangent are well defined only BECAUSE all right triangles with a given acute angle are similar — the ratios depend on the angle and not on the size.
The square-cube law
Lengths scale by k, areas by k², volumes by k³. That mismatch has consequences well beyond geometry.
Double an animal in every dimension: its weight goes up eightfold while the cross-section of its bones goes up only fourfold. The stress on them doubles. This is why large animals have proportionally thicker limbs, and why a scaled-up insect could not stand.
It is why small creatures survive falls that would kill larger ones — air resistance goes with area, weight with volume. And it is why heat loss is a problem for small animals and cooling is a problem for large machines.
All of it follows from the same fact this page reports: scale by k, and area goes by k².
Sources and methodology
The similarity criteria and the scaling laws are standard; these are the references.
Method. The scale factor comes from the first complete pair. When you supply more than one pair, all the ratios are compared and similarity is reported as verified or refuted rather than assumed — two pairs with different ratios mean the triangles are not similar, which is a more useful answer than a factor. The area factor is derived as the square of the length factor rather than computed separately, so the two cannot disagree. That engine is verified on every change against 76 hand-written assertions, including that the area factor is exactly the square of the length factor across two thousand generated pairs, and that a completed side always equals its original times that factor. The count and the per-case breakdown are published on the formula verification page.
Read the guide
A similar-triangle problem usually reduces to solving a : b = c : x, and the Proportion Calculator does that step with the cross multiplication shown.