Similar Triangles Calculator
Check whether two triangles are congruent or similar from their sides, find missing sides from a scale factor, use the basic proportionality theorem, and compare areas of similar triangles.
About the Similar Triangles Calculator
Two triangles are similar when their angles are equal, which happens exactly when their matching sides are in one ratio (SSS similarity), two sides are in proportion with the included angle equal (SAS), or two angles match (AA). They are congruent when the ratio is 1.
Thales' basic proportionality theorem: a line parallel to one side of a triangle divides the other two sides in the same ratio. The areas of similar triangles are in the ratio of the squares of matching sides.
The formula
AB/PQ = BC/QR = CA/RP = k; area ratio = k²; AD/DB = AE/EC
Worked example
Example: The triangles are = Similar (SSS), k = 2
- Inputs: I want: Are two triangles congruent or similar? (three sides each); Sides of triangle ABC (AB, BC, CA): 5, 12, 13; Sides of triangle PQR (PQ, QR, RP): 10, 24, 26.
- Formula: AB/PQ = BC/QR = CA/RP = k; area ratio = k²; AD/DB = AE/EC
- Ratio of the shortest sides: (10)/(5) = (2)
- Ratio of the middle sides: (24)/(12) = (2)
- Ratio of the longest sides: (26)/(13) = (2)
- Result: The triangles are = Similar (SSS), k = 2
- Side ratios (shortest to longest): 2, 2, 2
- Area ratio: 4
Things to keep in mind
- Congruence by SAS, ASA, AAS and RHS needs angles; this page tests sides (SSS). Use the triangle calculator to find the angles first.
Questions people ask
Is congruent the same as similar?
Congruent triangles are similar with scale factor 1: same shape and same size.
Why squares for areas?
Area is base × height ÷ 2 and both lengths scale by k, so area scales by k².