Check Similarity in Right Triangles Calculator
Enter any two sides of each right triangle and this calculator tells you instantly whether they are similar, what the scale factor is, and whether a reflection is involved. You also get all three side lengths for both triangles, the full ratio proof, and a show-your-work panel so you can follow the math at every step.
Formula
Worked example
Triangle 1 has legs 3 and 4 (hypotenuse 5). Triangle 2 has legs 6 and 8 (hypotenuse 10). Ratios: 6/3 = 2, 8/4 = 2, 10/5 = 2. All three ratios are equal, so the triangles are similar with scale factor k = 2.
What does it mean for two right triangles to be similar?
Two right triangles are similar if they have exactly the same shape, regardless of size. Because both triangles already share a right angle (90 degrees), you only need to confirm that the two acute angles match as well. The quickest numerical test is to check that all three pairs of corresponding sides share the same ratio, called the scale factor. If the short leg, long leg, and hypotenuse of Triangle 2 are all exactly k times the corresponding sides of Triangle 1, the triangles are similar with scale factor k.
Similarity is different from congruence: congruent triangles are identical in both shape and size (scale factor = 1), while similar triangles share the same shape but may differ in size. Enlarging a photo, drawing a blueprint, or scaling a map all produce similar triangles.
How the calculator checks similarity
The calculator first resolves all three sides of each triangle using the Pythagorean theorem (c = sqrt(a^2 + b^2)), then sorts the sides from smallest to largest so corresponding sides are always compared correctly. It then divides each side of Triangle 2 by the matching side of Triangle 1 and checks whether the three resulting ratios are equal (within a small floating-point tolerance). If they are, the triangles are similar and the common ratio is the scale factor.
If the direct ratios differ, the calculator tries the swapped correspondence: it checks whether Triangle 2 is a reflected (mirror-image) version of Triangle 1. This happens when the legs are interchanged, which is geometrically meaningful for non-isosceles right triangles because flipping a triangle is a valid similarity transformation. The "reflection needed" output tells you whether this extra step is required to align the two triangles.
The three similarity criteria for triangles
Mathematicians recognise three equivalent criteria for triangle similarity:
- AA (Angle-Angle): If two pairs of angles are equal, the third must be too (angles sum to 180 degrees). For right triangles, the right angle is given, so one equal acute angle is enough to confirm AA similarity.
- SSS (Side-Side-Side): All three corresponding side ratios are equal. This is the numerical check this calculator performs directly.
- SAS (Side-Angle-Side): Two side ratios equal and the angle between them equal. In a right triangle the right angle is always "between" the two legs, so equal leg ratios automatically satisfy SAS as well.
Worked example and common Pythagorean triples
The classic example: a 3-4-5 triangle and a 6-8-10 triangle. Dividing each side of the larger by the corresponding side of the smaller gives 6/3 = 8/4 = 10/5 = 2, confirming similarity with scale factor 2. The larger triangle is a scaled-up copy of the smaller one.
Common Pythagorean triples - integer side lengths that form right triangles - include:
- 3-4-5 (and multiples: 6-8-10, 9-12-15, ...)
- 5-12-13
- 8-15-17
- 7-24-25
Triangle similarity criteria
| Criterion | What to check | Right triangle shortcut |
|---|---|---|
| AA (Angle-Angle) | Two pairs of equal angles | One right angle + one equal acute angle |
| SSS (Side-Side-Side) | All three side ratios equal | Both leg ratios equal (hypotenuse follows) |
| SAS (Side-Angle-Side) | Two side ratios equal and the included angle equal | Leg ratio equal + right angle (always met) |
The three standard criteria for deciding whether two triangles are similar.
Frequently asked questions
How do I check if two right triangles are similar?
Compute all three sides of each triangle using the Pythagorean theorem, then check whether the ratios of corresponding sides are all equal. If short-leg1/short-leg2 = long-leg1/long-leg2 = hyp1/hyp2, the triangles are similar. Alternatively, check that the acute angle (or equivalently the leg ratio a/b) is the same in both triangles.
What is the scale factor of similar triangles?
The scale factor k is the constant ratio of corresponding sides: k = (any side of Triangle 2) / (the matching side of Triangle 1). For similar triangles all three ratios are identical. If k > 1 then Triangle 2 is larger; if k < 1 it is smaller; if k = 1 the triangles are congruent.
Are a 3-4-5 triangle and a 6-8-10 triangle similar?
Yes. Dividing each side of the 6-8-10 triangle by the corresponding side of the 3-4-5 triangle gives 6/3 = 2, 8/4 = 2, 10/5 = 2. All three ratios are equal, so the triangles are similar with scale factor 2.
What does "reflection needed" mean?
A reflection (mirror flip) is needed when the two triangles are similar in shape but one is oriented as the mirror image of the other. This happens when Triangle 2 has a short leg where Triangle 1 has a long leg. After flipping Triangle 2, the correspondence short-to-short and long-to-long holds and the scale factor can be read off normally.
Do two right triangles always have the same angles?
No. All right triangles share the 90-degree angle, but their two acute angles depend on the leg ratio. A 1-1-sqrt(2) triangle has two 45-degree angles, while a 3-4-5 triangle has angles of approximately 36.87 and 53.13 degrees. Only triangles with the same leg ratio (and therefore the same acute angles) are similar.
Can two right triangles be similar without knowing all sides?
Yes. Because both triangles already have a right angle, you only need one additional pair of equal angles (AA criterion) to confirm similarity. If you know one acute angle of each triangle and they match, the triangles are similar regardless of their size.
How do I find a missing side in a similar right triangle?
Identify the scale factor k from the sides you know, then multiply or divide. If Triangle 1 has sides 3, 4, 5 and Triangle 2 is similar with scale factor 3, its sides are 9, 12, 15. If the scale factor is unknown but you know two sides of Triangle 2, divide each by the matching side of Triangle 1 to find k, then apply k to all remaining sides.