Triangle Congruence Calculator
Enter three side lengths for two triangles. The calculator computes all angles, checks SSS, SAS, ASA, AAS, and HL congruence, and tells you whether the triangles are congruent, merely similar, or unrelated.
Triangle 1 — sides
Triangle 2 — sides
Triangles are congruent: all corresponding sides and angles are equal.
- 1
Triangle 1 semi-perimeter
(3 + 4 + 5) ÷ 2 = 6 - 2
Heron's product
6 × 3 × 2 × 1 = 36The product inside Heron's formula √[s(s−a)(s−b)(s−c)]. - 3
Triangle 1 area
√36 = 6
How does this calculator work?
Enter three sides for each triangle. If all sorted side lengths match (SSS), the triangles are congruent — and SSS, SAS, ASA, AAS all hold simultaneously. For right triangles, HL (hypotenuse + one leg equal) also applies. If only angles match the triangles are merely similar (same shape, different size).
Formula
How this is calculated
Two triangles are congruent if they have the same size and shape — every corresponding side and every corresponding angle is equal. The most fundamental congruence criterion is SSS (Side-Side-Side): if all three pairs of corresponding sides are equal, the triangles are congruent. Because the Law of Cosines uniquely determines all angles from three sides, satisfying SSS automatically satisfies the other geometric criteria — SAS (two sides and the included angle), ASA (two angles and the included side), and AAS (two angles and a non-included side) — as all these follow from the same set of side lengths.
This calculator works by taking the three side lengths of each triangle, computing all three interior angles via the Law of Cosines (cos A = (b²+c²−a²)/(2bc)), and then comparing the sorted side lists and sorted angle lists. Sorting means you can enter the sides in any order — the calculator matches the shortest to shortest, etc. For right triangles, the additional HL (Hypotenuse-Leg) criterion is checked: if both triangles are right triangles and have matching hypotenuse and one leg, they are congruent.
If the angles match but the sides differ, the triangles are similar — same shape, different scale. If neither sides nor angles match, the triangles are geometrically unrelated. The calculator flags each case and lists all applicable congruence criteria.
Frequently asked questions
Congruent triangles are identical in size and shape — all corresponding sides and all corresponding angles are equal. Similar triangles have the same angle measures (same shape) but proportionally different side lengths (different size). All congruent triangles are similar, but not all similar triangles are congruent.
Given three specific side lengths, the triangle inequality ensures a unique triangle exists (up to reflection and rotation). The Law of Cosines then uniquely determines all angles. So there is only one triangle (up to flipping) that has those three side lengths, proving SSS congruence.
HL (Hypotenuse-Leg) is a special congruence criterion for right triangles. If two right triangles share the same hypotenuse length and the same leg length, they are congruent. It is a special case of SSS once you recover the missing leg via the Pythagorean theorem: missing leg = √(hypotenuse² − leg²).
Also known as
TG we-Calculate Editorial Team. (2026). Triangle Congruence Calculator [Online calculator]. TG we-Calculate. https://we-calculate.com/calculator/triangle-congruence-calculator
TG we-Calculate Editorial Team. "Triangle Congruence Calculator." TG we-Calculate. 2026. https://we-calculate.com/calculator/triangle-congruence-calculator.
TG we-Calculate Editorial Team, "Triangle Congruence Calculator," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/calculator/triangle-congruence-calculator
@misc{wecalculate_triangle_congruence_calculator, title = {Triangle Congruence Calculator}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/calculator/triangle-congruence-calculator}}, year = {2026}, note = {TG we-Calculate} }
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