How this instrument works
Two right triangles are similar — identical in shape, differing only in size — exactly when the ratio of their corresponding legs matches. This calculator computes leg a divided by leg b for each triangle separately, then subtracts the two ratios: a result of exactly 0 means the ratios match and the triangles are similar; any other result means they aren't.
This works because similarity means one triangle is a uniformly SCALED copy of the other — every length multiplied by the same scale factor. If triangle 2's legs are both triangle 1's legs multiplied by some factor k, then triangle 2's own leg ratio, (k·a₁) ⁄ (k·b₁), simplifies right back down to a₁ ⁄ b₁ — the scale factor cancels out entirely, leaving the ratio unchanged. Comparing ratios directly sidesteps ever needing to find that scale factor explicitly.
For right triangles specifically, matching leg ratios is enough on its own to guarantee similarity, since the right angle is already shared and fixed — once the ratio of the two legs matches, the third angle (determined by that ratio through basic trigonometry) must match too, making all three angles identical between the two triangles.
- Enter the first triangle's two legs into the Triangle 1: leg a and Triangle 1: leg b fields.
- Enter the second triangle's two legs into the Triangle 2: leg a and Triangle 2: leg b fields.
- Read the test value: exactly 0 means the two triangles are similar; any other value means they aren't.
Worked example — legs 3,4 versus legs 6,8
Right triangle 1 has legs 3 and 4; right triangle 2 has legs 6 and 8. The test value is (3⁄4) − (6⁄8) = 0.75 − 0.75 = 0 — SIMILAR, since triangle 2 is exactly triangle 1 with every length doubled, a uniform scaling that leaves the leg ratio completely unchanged.
Compare triangle 1 (legs 3, 4) against a triangle with legs 6 and 9 instead: (3⁄4) − (6⁄9) ≈ 0.75 − 0.667 ≈ 0.083, clearly nonzero — NOT similar, since the second triangle's legs don't scale by the same uniform factor, despite the numbers looking superficially close to the golden example.
Questions
How do you check if two right triangles are similar?
Compare the ratio of their legs: divide one triangle's leg a by its leg b, do the same for the second triangle, and check whether the two ratios match. Matching ratios mean the triangles are similar; different ratios mean they aren't.
Why is checking leg ratios enough for right triangles?
Because a right triangle already has one angle fixed at 90°. Once the ratio of the two legs matches between two right triangles, basic trigonometry forces the remaining angle to match too, guaranteeing all three angles — and therefore the full shape — are identical.
What does 'similar' mean geometrically?
Two shapes are similar when one is a uniformly scaled copy of the other — every corresponding length multiplied by the same scale factor, with every corresponding angle staying exactly equal. Similar triangles have identical shape but not necessarily identical size.
Does it matter which leg I call 'a' and which I call 'b'?
It has to stay CONSISTENT between the two triangles — whichever leg is called 'a' in the first triangle must correspond to the same relative position (the matching side) called 'a' in the second, or the ratio comparison won't test the right relationship.
How is this different from congruence?
Similarity only requires matching SHAPE (proportional sides, equal angles); congruence requires matching shape AND size exactly. Two right triangles can easily be similar without being congruent, as the golden example's doubled triangle shows.