Triangle Similarity Checker – Verify AA, SSS, and SAS Similarity
Two triangles are similar when they have the same shape but not necessarily the same size — every angle in one triangle matches a corresponding angle in the other, and every side is scaled by the same constant factor. This calculator checks two triangles against the three standard similarity theorems and reports the result along with the scale factor, the corresponding side ratios, and a plain-English explanation of why the triangles are (or are not) similar.
The Three Similarity Criteria
- AA (Angle-Angle)— if two pairs of corresponding angles are equal, the triangles are similar. Because a triangle's angles always sum to 180°, the third pair matches automatically.
- SSS (Side-Side-Side) — if all three pairs of corresponding sides are proportional, the triangles are similar. This is verified by sorting both side sets and checking that
d/a = e/b = f/c. - SAS (Side-Angle-Side) — if two pairs of sides are proportional and the angle between them is equal, the triangles are similar.
The checker tests SSS first (the most direct proof when all six sides are known), then AA, then SAS, and reports the first criterion that confirms similarity.
How the Scale Factor Is Calculated
The scale factor is the constant ratio between corresponding sides, expressed as Triangle 2 ÷ Triangle 1. For example, if a = 3, b = 4, c = 5 and d = 6, e = 8, f = 10, every ratio equals 2, so the triangles are similar by SSS with a scale factor of 2 — triangle 2 is exactly twice the size of triangle 1.
Deriving Missing Angles
You only need to supply the three side lengths of each triangle — angles are derived automatically using the law of cosines:
cos(A) = (b² + c² − a²) / (2bc)
This lets the AA check run even when no angles were entered directly, since the angles are computed from the sides before any comparison is made.
Handling Mixed Units
Each triangle has its own unit selector (mm, cm, m, in, ft). Before any ratio is computed, both triangles are converted to a common base unit internally, so you can safely compare a triangle measured in inches against one measured in centimeters without manual conversion.
When Triangles Are Not Similar
If none of the three criteria hold, the calculator reports Not Similar and shows the individual side ratios so you can see exactly where the proportionality breaks down. Before running any similarity test, each triangle is validated against the triangle inequality — the sum of any two sides must exceed the third — to catch impossible triangle configurations early.
Practical Applications
Triangle similarity is foundational in geometry education, and it also has practical uses: architects and engineers rely on similar triangles when scaling blueprints and structural models, surveyors use similarity to measure distances indirectly, and artists and designers use proportional scaling to enlarge or shrink shapes while preserving their form.
Using This Calculator
Enter the three side lengths for Triangle 1 and Triangle 2, choose the length unit for each, and the result updates instantly. The result panel shows which similarity rule applies, the scale factor, the individual side ratios, and the derived angles for both triangles, alongside a labeled diagram of each triangle drawn to scale. Use the copy button to export the full result as plain text.