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Triangle Congruence Calculator

Determine if two triangles are congruent using the SSS (Side-Side-Side) test. Enter the three side lengths of each triangle to check if they match when sorted, confirming the triangles are identical in shape and size for geometry proofs and engineering.

Reviewed by Christopher FloiedUpdated

This free online triangle congruence calculator provides instant results with no signup required. All calculations run directly in your browser — your data is never sent to a server. Enter your values below and see results update in real time as you type. Perfect for everyday calculations, homework, or professional use.

First side of triangle 1

Second side of triangle 1

Third side of triangle 1

First side of triangle 2

Second side of triangle 2

Third side of triangle 2

Results

Congruent by SSS (1 = Yes, 0 = No)

1

How to Use This Calculator

1

Enter your input values

Fill in all required input fields for the Triangle Congruence Calculator. Most fields include unit selectors so you can work in your preferred unit system — metric or imperial, whichever matches your problem.

2

Review your inputs

Double-check that all values are correct and that you have selected the right units for each field. Incorrect units are the most common source of calculation errors and can produce results that are off by factors of 2, 10, or more.

3

Read the results

The Triangle Congruence Calculator instantly computes the output and displays results with units clearly labeled. All calculations happen in your browser — no loading time and no data sent to a server.

4

Explore parameter sensitivity

Try adjusting individual input values to see how the output changes. This is a quick and effective way to develop intuition about how different parameters influence the result and to identify which inputs have the largest effect.

Formula Reference

Triangle Congruence Calculator Formula

See calculator inputs for the governing equation

Variables: All variables and their units are labeled in the calculator interface above. Input fields accept values in multiple unit systems — select your preferred unit from the dropdown next to each field.

When to Use This Calculator

  • •Use the Triangle Congruence Calculator when you need a quick mathematical result without writing out all the steps manually, saving time on repetitive calculations.
  • •Use it to verify hand calculations on tests or assignments and catch arithmetic mistakes.
  • •Use it when teaching or explaining mathematical concepts to others, demonstrating how changing inputs affects the result.
  • •Use it to explore the behavior of mathematical functions across a range of inputs.

About This Calculator

The Triangle Congruence Calculator is a free mathematical calculation tool for students, educators, and professionals who need quick, reliable results. Determine if two triangles are congruent using the SSS (Side-Side-Side) test. Enter the three side lengths of each triangle to check if they match when sorted, confirming the triangles are identical in shape and size for geometry proofs and engineering. The underlying algorithms implement well-established mathematical formulas and numerical methods. Results are computed instantly in the browser. This tool is useful for learning, verification of hand calculations, and rapid exploration of mathematical relationships. All computation happens locally — no data is sent to a server.

About Triangle Congruence Calculator

The Triangle Congruence Calculator determines whether two triangles are congruent using the Side-Side-Side (SSS) criterion. Two triangles are congruent if and only if their corresponding sides are equal in length. The calculator sorts the sides of each triangle and checks if the smallest, middle, and largest sides match. This is the most fundamental congruence test in Euclidean geometry. Congruence testing is used in engineering (verifying manufactured parts match specifications), computer graphics (mesh deduplication), structural analysis (ensuring symmetric components), and education (geometry proofs). The SSS test is one of five congruence criteria: SSS, SAS, ASA, AAS, and HL.

The Math Behind It

Two triangles are congruent if there is a correspondence between their vertices such that all three pairs of corresponding sides are equal and all three pairs of corresponding angles are equal. Remarkably, you do not need to verify all six conditions. Euclid proved that three conditions suffice, depending on which elements are specified. SSS (Side-Side-Side): If all three sides of one triangle equal the corresponding sides of another, the triangles are congruent. This was Proposition 8 in Book I of Euclid's Elements. The proof uses the method of superposition: place one triangle on the other so two sides coincide, and show the third sides must also coincide. Other congruence criteria include: SAS (two sides and the included angle), ASA (two angles and the included side), AAS (two angles and a non-included side), and HL (hypotenuse-leg for right triangles). Note that SSA (two sides and a non-included angle) is NOT generally valid, as it can produce two different triangles (the ambiguous case). Congruence is an equivalence relation: it is reflexive (every triangle is congruent to itself), symmetric (if A is congruent to B, then B is congruent to A), and transitive (if A is congruent to B and B is congruent to C, then A is congruent to C). In coordinate geometry, congruent triangles are related by rigid motions: translations, rotations, and reflections. In modern mathematics, congruence of triangles is a special case of isometry between metric spaces. The rigidity of triangles (determined by their three sides) contrasts with quadrilaterals, which are not rigid (a rectangle can be deformed into a parallelogram without changing side lengths).

Formula Reference

SSS Congruence Test

Triangle 1 congruent to Triangle 2 iff sorted sides match

Variables: Three side lengths from each triangle, compared after sorting

Worked Examples

Example 1: Congruent 3-4-5 Triangles

Triangle 1 has sides 3, 4, 5. Triangle 2 has sides 5, 3, 4.

Step 1:Sort Triangle 1 sides: 3, 4, 5
Step 2:Sort Triangle 2 sides: 3, 4, 5
Step 3:Compare: min(3,4,5) = 3 = min(5,3,4) = 3 (match)
Step 4:max(3,4,5) = 5 = max(5,3,4) = 5 (match)
Step 5:Sum: 12 = 12 (match, confirms middle sides equal too)

The triangles are congruent by SSS.

Example 2: Non-Congruent Triangles

Triangle 1 has sides 3, 4, 5. Triangle 2 has sides 3, 4, 6.

Step 1:Sort both: T1 = (3,4,5), T2 = (3,4,6)
Step 2:Smallest sides match: 3 = 3
Step 3:Largest sides differ: 5 != 6

The triangles are NOT congruent.

Common Mistakes & Tips

  • !Comparing sides in the original given order without sorting. Side a of triangle 1 might correspond to side c of triangle 2.
  • !Confusing congruence with similarity. Congruent triangles have equal sides; similar triangles have proportional sides with equal angles.
  • !Assuming SSA (two sides and a non-included angle) is a valid congruence test. SSA can produce two different triangles (the ambiguous case).

Related Concepts

Triangle Similarity

Similar triangles have proportional sides and equal angles. Congruence is the special case where the scale factor is 1.

Triangle Inequality

Before testing congruence, verify that the side lengths actually form valid triangles using the triangle inequality.

Used in These Calculators

Calculators that build on or apply the concepts from this page:

Triangle Inequality Calculator

Triangle Similarity Calculator

Frequently Asked Questions

What are all the triangle congruence criteria?

The five valid criteria are: SSS (three sides), SAS (two sides and the included angle), ASA (two angles and the included side), AAS (two angles and a non-included side), and HL (hypotenuse-leg for right triangles). SSA is NOT valid in general due to the ambiguous case.

Why is SSA not a valid congruence test?

Because given two sides and a non-included angle, there can be zero, one, or two possible triangles. This ambiguity means you cannot uniquely determine the triangle, so congruence is not guaranteed.

What is the difference between congruence and equality?

Congruent triangles have the same shape and size but may differ in position and orientation. They are related by rigid motions (translations, rotations, reflections). Equality typically refers to identical objects in the same position.