Similar Triangles Calculator
Enter the three sides of triangle A and one side of triangle B to find all missing sides, the scale factor, area ratio, and perimeter ratio between similar triangles.
Congruent Triangles Calculator
Enter the sides and angles of two triangles to check congruence. Tests SSS, SAS, ASA, AAS, and HL criteria, identifies which criterion matched, solves missing values, and compares areas.
Triangle 1
Triangle 2
Planning notes, formulas, and examples
About the Congruent Triangles Calculator
Two triangles are congruent when they have exactly the same shape and size — every corresponding side is equal and every corresponding angle is equal. Proving congruence does not require checking all six measurements; instead, five classic criteria provide shortcuts: SSS (three pairs of equal 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).
Congruence is fundamental in Euclidean geometry, architecture, and engineering. Structural trusses rely on congruent triangles for uniform load distribution. CNC machining checks that cut parts are congruent to the master template. In academic settings, students must identify the correct criterion in proof-based problems.
This calculator lets you enter up to three sides and three angles for each triangle, then automatically tests every applicable criterion and reports which one (if any) confirms congruence. It also solves for any missing values using the law of cosines and the angle-sum property, compares both areas, and displays a visual side-by-side bar chart. Presets load common textbook examples so you can explore criteria quickly without manual entry. A reference table summarizes all five criteria with their requirements.
When This Page Helps
Use this when you need to test whether two triangles match exactly under SSS, SAS, ASA, AAS, or HL conditions without manually checking every congruence rule. It is helpful for proofs and exam practice because the side-angle evidence and the congruence verdict come from the same pair of triangles.
How to Use the Inputs
- Enter the known sides (a, b, c) and/or angles (A, B, C) for Triangle 1.
- Enter the known sides (a, b, c) and/or angles (A, B, C) for Triangle 2.
- Click a preset to load a classic congruence example.
- The calculator tests each criterion (SSS, SAS, ASA, AAS, HL) and reports which matched.
- Missing sides and angles are solved automatically using the law of cosines and 180° rule.
- Compare the areas and perimeters side by side in the output cards.
- View the visual bar chart for a quick size comparison of corresponding sides.
Formula used
SSS: a₁ = a₂, b₁ = b ₂, c₁ = c₂
SAS: two sides equal and included angle equal
ASA: two angles equal and included side equal
AAS: two angles equal and a non-included side equal
HL: right triangle with hypotenuse and one leg equal
Law of cosines: c² = a² + b² − 2ab·cos(C)
Angle sum: A + B + C = 180°Example Calculation
Result: For a1=3, b1=4, c1=5, the tool returns the solved congruent triangles outputs shown in the result cards.
This example uses a realistic input set from the calculator workflow. After entry, the calculator applies the built-in congruent triangles formulas and reports derived values, checks, and classifications automatically.
Tips & Best Practices
- SSS is the most straightforward test — measure all three sides of each triangle.
- For SAS, the angle must be between the two known sides (the included angle).
- ASA and AAS both involve two angles, but differ in which side is known relative to them.
- HL only applies to right triangles — make sure one angle is exactly 90°.
- Congruent triangles always have equal areas and perimeters.
- SSA (side-side-angle where the angle is not included) is NOT a valid criterion — it can produce ambiguous cases.
When To Use This Calculator
Enter the sides and angles of two triangles to check congruence. Tests SSS, SAS, ASA, AAS, and HL criteria, identifies which criterion matched, solves missing values, and compares areas. Use it when you need a repeatable calculation in the math / geometry category and want the setup, result, and supporting values kept together. This is especially helpful when small input changes, unit choices, or rounding decisions can change the final number.
How To Check The Result
Start by confirming that the inputs match the formula shown on the page. Then compare the main output with the worked example and any secondary values shown by the calculator. If the result will be used in another calculation, keep extra precision until the final step and record the assumptions beside the number.
Practical Notes
Treat the result as a calculation aid rather than a substitute for context. For schoolwork, include the formula and substitution steps. For planning, technical, financial, or health-related decisions, verify important numbers against primary records, current rules, or a qualified professional before acting on them.
Sources & Methodology
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Frequently Asked Questions
Congruent triangles are identical in shape AND size (all sides and angles equal). Similar triangles have the same shape but may differ in size (angles equal, sides proportional).
SSA (two sides and a non-included angle) can produce two different triangles (the ambiguous case), so it does not guarantee a unique triangle.
HL stands for Hypotenuse-Leg. It applies only to right triangles: if the hypotenuse and one leg of one right triangle equal those of another, the triangles are congruent.
You need exactly three correctly chosen measurements for each triangle — matching one of the five criteria (SSS, SAS, ASA, AAS, or HL).
Yes. Congruent triangles can be reflections (mirror images) of each other. Congruence requires equal corresponding parts, regardless of orientation.
Yes. Since all corresponding sides and angles are equal, congruent triangles always have identical area, perimeter, and every other measurement.
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