Check Similarity in Right Triangles Calculator

Enter the two legs of two right triangles to test similarity with the SAS Similarity Theorem — get each hypotenuse, the scale factor, and a clear similar/not-similar verdict.

Quick Facts

Similarity rule
SAS Similarity Theorem
Every right angle is 90°, so if each triangle's two legs are in the same ratio, the triangles are similar.
Scale factor
k = corresponding side ÷ corresponding side
When similar, legs and hypotenuse are all scaled by the same factor k.

Your Results

Calculated
Hypotenuse — Triangle 1
-
c₁ = √(a₁² + b₁²)
Hypotenuse — Triangle 2
-
c₂ = √(a₂² + b₂²)
Scale factor (k)
-
Average of a₂/a₁ and b₂/b₁
Leg-ratio difference
-
0% means a perfect match

Ready

Enter both triangles' legs, then press Calculate.

How the Right Triangle Similarity Check works

This calculator tests whether two right triangles are similar using the SAS (Side-Angle-Side) Similarity Theorem. Every right triangle already contains a 90° angle, so that one angle is automatically congruent between any two right triangles. What remains is to check whether the two sides that form that right angle — the legs — are in the same ratio. If they are, the triangles are similar; if not, they are not, no matter how close the shapes look.

Formula and method

For Triangle 1 with legs a₁, b₁ and Triangle 2 with legs a₂, b₂, the calculator computes:

  • Hypotenuse of each triangle (Pythagorean theorem): c₁ = √(a₁² + b₁²) and c₂ = √(a₂² + b₂²)
  • Leg ratios: k₁ = a₂ ÷ a₁ and k₂ = b₂ ÷ b₁
  • Similarity test: if k₁ and k₂ agree (within a 0.5% tolerance for rounding), the triangles are similar by SAS, with scale factor k equal to their average

Because the hypotenuse is fixed by the two legs (Pythagorean theorem), a true scale factor k also satisfies c₂ = k × c₁ automatically — so the hypotenuse ratio serves as a built-in cross-check on the leg-ratio test.

Common sources of error

  • Mismatched leg order: "Leg a" in Triangle 1 must correspond to "Leg a" in Triangle 2 (and likewise for Leg b). Swapping which leg goes where can make genuinely similar triangles look dissimilar.
  • Entering the hypotenuse as a leg: only the two sides that form the right angle go in the leg fields — never the longest side.
  • Over-trusting tiny mismatches: real measurements rarely produce a perfect ratio, so this tool allows up to 0.5% difference between the two leg ratios before calling the triangles "not similar."

Checking your result

A quick sanity check: if you can spot a common multiplier between the two triangles' legs (for example, one triangle's legs are exactly double the other's), the scale factor k reported here should match that multiplier, and the leg-ratio difference should read 0%.

Applications

Similar right triangles come up in scale drawings and blueprints, indirect measurement (using a stick's shadow and a building's shadow, which form similar right triangles), map and model scaling, and geometry proofs — including the altitude-to-the-hypotenuse construction, where dropping a perpendicular from the right angle creates two smaller right triangles that are similar to the original and to each other.

Frequently Asked Questions

How do you check if two right triangles are similar?
Compare the ratio of corresponding legs: divide Triangle 2's first leg by Triangle 1's first leg (a₂/a₁), and do the same for the second legs (b₂/b₁). If both ratios come out equal, the right angle plus the proportional legs satisfy the SAS Similarity Theorem, so the triangles are similar.
What is the SAS Similarity Theorem for right triangles?
SAS (Side-Angle-Side) Similarity says two triangles are similar if one angle in each triangle is congruent and the two sides forming that angle are proportional. Every right triangle already has a 90° angle, so comparing the two legs — the sides that form the right angle — is enough to test similarity.
What does the scale factor mean?
The scale factor k tells you how many times larger or smaller one triangle is than the other. If k = 2, every side of the second triangle — both legs and the hypotenuse — is exactly twice the length of the matching side in the first triangle.
Why does the hypotenuse ratio also match when triangles are similar?
By the Pythagorean theorem, a right triangle's hypotenuse is determined by its two legs. If both legs are scaled by the same factor k, the hypotenuse scales by k too (c₂ = k × c₁), so a genuine match in the leg ratios automatically carries over to the hypotenuse ratio.

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