How to Tell if Two Triangles Are Similar

Geometry Lessons

The short answer

every side ratio the same

Sort both triangles, pair the sides in order, and divide. If all three ratios match, the triangles are similar. Two equal angles is enough on its own.

Similar means same shape, different size. The angles are identical and every length has been multiplied by one constant, called the scale factor. It is the relationship behind map scales, model drawings and anything measured from a photograph, so it earns its keep well outside a geometry class.

Three ways to prove it

The quickest test is two equal angles. Since the three angles of a triangle always sum to 180, fixing two fixes the third automatically, so two matching angles guarantee all three match. That is the AA test, and it needs no side lengths at all.

If you have lengths rather than angles, compare all three ratios. This is the SSS test, and it is the one to use on measured data. If you have two sides and the angle sitting between them, matching ratios on those two sides plus an equal included angle is enough, which is the SAS test.

What never works is comparing sides that do not correspond. Similarity is about matching each side to its counterpart, and the safe way to do that is to sort both triangles by length before pairing anything up.

1

Sort both triangles

Smallest to largest. This pairs each side with its counterpart without guessing.

2

Divide each pair

Take the second triangle over the first, three times.

3

Check all three agree

One constant ratio means similar. Any disagreement means they are not.

A worked example

Take a 3, 4, 5 triangle and a 6, 8, 10 triangle. Both are already sorted, so pair them straight across: 3 with 6, 4 with 8, and 5 with 10. Dividing gives 2, 2 and 2. All three agree, so the triangles are similar with a scale factor of 2.

Now change the second triangle to 6, 8, 11. The ratios become 2, 2 and 2.2. Two out of three matching is not enough. Those triangles are genuinely different shapes, and the third angle differs even though the first two sides scaled cleanly.

A small right triangle with sides three four five drawn beside a larger one with sides six eight ten on the same baseline, with the working showing every side ratio equal to two, the perimeter doubling and the area quadrupling.
The perimeter scales by the same factor as the sides, but the area does not. That difference is where most of the real mistakes with similarity happen.

Areas scale by the square, not the factor

This is the error that costs real money rather than marks. Doubling the sides of a triangular area does not double the material needed, it quadruples it. The sides and the perimeter scale by the factor, and the area scales by its square.

scale 2givesarea 6 becomes 12wrong

treating area as if it scaled like a length

scale 2givesarea 6 becomes 244 times

the factor squared, since area covers two dimensions

Where the scale factor gets used

Any scaled drawing is an exercise in similarity. A plan at 1 to 50 produces triangles similar to the real ones, so a length measured on paper multiplied by 50 gives the length on site. The angles are unchanged, which is why you can read a slope or a corner straight off a drawing without correction.

The area trap follows immediately. A shape covering 4 square centimetres on a 1 to 50 plan does not cover 200 square centimetres on the ground. It covers 4 times 50 squared, which is 10,000 square centimetres, or 1 square metre. Getting this wrong on a takeoff is expensive.

Similar is not the same as congruent

The two words get used loosely and they mean different things. Similar allows any scale factor, so the shapes match but the sizes need not. Congruent requires a scale factor of exactly 1, meaning identical size as well as shape.

Every congruent pair is also similar. The reverse is not true, which is why proving all three angles equal shows similarity but never congruence.

Mistakes that catch people out

  • Pairing sides that do not correspond. Sort both triangles first so each side meets its counterpart.
  • Scaling area by the factor. Area scales by the factor squared. Doubling the sides quadruples the area.
  • Accepting two matching ratios. All three must agree. Two out of three is a different shape.
  • Confusing similar with congruent. Similar allows any size, congruent requires identical size.
  • Using AAA to claim congruence. Three equal angles proves similarity only, and says nothing about size.

Practice questions

  1. A 4, 6, 8 triangle and a 6, 9, 12 triangle. Are they similar?
  2. A 5, 12, 13 triangle and a 10, 24, 26 triangle. Are they similar?
  3. A 3, 4, 5 triangle and a 6, 8, 11 triangle. Are they similar?
  4. Two similar triangles with a scale factor of 3. The smaller has area 7. Find the larger area.
  5. Two similar triangles have areas of 5 and 45. Find the scale factor.
Show the answers
  1. ratios 1.5, 1.5 and 1.5, so yes, with a scale factor of 1.5
  2. ratios 2, 2 and 2, so yes, with a scale factor of 2
  3. ratios 2, 2 and 2.2, so no, the third ratio breaks it
  4. area scales by 3 squared, so 7 x 9 = 63
  5. the area ratio is 9, so the scale factor is its square root, which is 3

Common questions

How do I know if two triangles are similar?

Sort both by side length, pair them in order and divide. If all three ratios are the same, they are similar. Alternatively, two matching angles is enough on its own.

Is two equal angles really enough?

Yes. The three angles sum to 180, so fixing two fixes the third. This is the AA test and it needs no side lengths.

Do similar triangles have to be the same size?

No, that is congruence. Similar triangles have the same shape at any size. Congruent triangles are similar with a scale factor of exactly 1.

How do areas scale?

By the square of the scale factor. Doubling every side multiplies the area by four, and tripling multiplies it by nine.

How do perimeters scale?

By the scale factor itself, the same as the sides. Only the area picks up the square.

What if only two of the three ratios match?

They are not similar. All three have to agree, and two matching is a common near miss that hides a genuinely different shape.

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