Triangle Similarity Calculator - AA, SSS, SAS Check

A triangle similarity calculator decides whether two triangles are similar under the AA, SSS, or SAS criterion and returns the scale factor, perimeter ratio, and area ratio in one pass, the same family of results you would reach by hand on a textbook problem about corresponding sides.

• • Homework and textbook problems: Test two given triangles and read the verdict, scale factor, and ratios in one step.
• • Shadows and indirect measurement: Compare a stick-and-shadow triangle with a tree-and-shadow triangle before solving for an unknown height.
• • Maps, models, and scale drawings: Confirm that a model triangle is similar to the real one, then read the scale factor that converts model lengths to real lengths.
• • Right-triangle families: Check whether a right triangle is in the 3-4-5, 5-12-13, or 8-15-17 family before solving for a missing side.

Two triangles are similar when their corresponding angles are equal and their corresponding sides are in the same ratio, so AA, SSS, and SAS are three entry points into the same conclusion. Pick the mode that matches your data: SSS for six side lengths, AA for six angle measurements, SAS for two side pairs and the included angle.

For the matching a/b = c/d proportion that solves a single missing side, Similar Triangles Calculator is the natural next step after the verdict and scale factor are known.

The triangle similarity calculator runs a single test based on the mode you pick. SSS divides the sides of triangle 2 by the matching sides of triangle 1 and checks whether the three ratios agree; AA checks whether two pairs of corresponding angles agree; SAS checks whether two side ratios agree and the included angle is the same. In every mode, k = a2 / a1 is the answer that links corresponding sides.

• a1, b1, c1: Three sides of the reference (smaller) triangle, used as divisors when the scale factor is computed.
• a2, b2, c2: Three sides of the second triangle, written in the same order as a1, b1, c1 so that a2 pairs with a1, b2 with b1, and c2 with c1.
• k: Linear scale factor from triangle 1 to triangle 2, equal to a2/a1 (or any other pair of corresponding sides) when the triangles are similar.
• tolerance: Maximum allowed relative deviation for side ratios and absolute deviation for angles, with a default of 0.01 to match how textbook problems are usually graded.

When the criterion fails, the verdict reads 'Not similar' and the reason line points to the disagreeing ratio or angle.

According to Wolfram MathWorld, two triangles are similar if and only if the ratios of their corresponding sides are equal, and the ratio of their areas is the square of that linear scale factor.

When the verdict says Similar and the proportion is the only thing left to use, Solve Similar Triangles rearranges the same a2/a1 = b2/b1 = c2/c1 ratio for any one missing side.

These four ideas decide how a similarity test is set up and how the scale factor, perimeter ratio, and area ratio are read from the result.

AA, SSS, and SAS are three different ways to reach the same conclusion. AA is the cheapest to measure with a protractor, SSS the most reliable with a ruler, and SAS the right tool when you have side-angle-side data from a CAD or surveying record.

Right triangles are a special case of similar triangles, so the 3-4-5, 5-12-13, and 8-15-17 families all share the same shape and Pythagorean Triples Calculator lists the common integer examples.

Pick the mode that matches the data you have, type the matching fields, and read the verdict, the reason line, the scale factor, and the two ratios in the results panel.

1. 1 Pick the similarity mode: Set the Similarity mode dropdown to SSS, AA, or SAS. SSS uses all six side fields; AA uses the six angle fields and one side pair; SAS uses two side pairs and the included angle.
2. 2 Adjust the tolerance if needed: Leave Tolerance at 0.01 for textbook problems. Tighten it to 0.001 for high-precision comparisons or relax it to 0.05 for hand-measured triangles.
3. 3 Enter the data for both triangles: Type the three sides of triangle 1 in a, b, c order, then the three sides of triangle 2 in the same order. AA mode also needs the six angles; SAS mode needs the two included angles.
4. 4 Read the verdict and the scale factor: The verdict line says Similar or Not similar, and k = a2 / a1 is the linear scale factor with the perimeter ratio equal to k and the area ratio equal to k squared.

A surveyor measures a 5 foot stick and a 3 foot shadow, then a 24 foot target shadow. The triangles are similar by AA, so k = 24 / 3 = 8 and the tree height is 5 * 8 = 40 feet.

When the second triangle in the test is a right triangle, Right Triangle Calculator extends the same side inputs to a full Pythagorean solution for the angles and the area.

Putting the AA, SSS, and SAS tests in one calculator keeps the similarity decision and the scale factor in the same result.

• • Three criteria in one place: Switch between AA, SSS, and SAS without rewriting the formula. The calculator reads whichever fields match the chosen mode.
• • Verdict plus a reason line: Get a Similar or Not similar verdict and a one-sentence reason that says which ratio or angle disagreed, the line you quote in a written solution.
• • Scale factor with no second pass: k = a2 / a1, the perimeter ratio, and the area ratio all appear at once.
• • Configurable tolerance: Adjust the tolerance to match how your lengths or angles were measured. Default 0.01 is fine for textbook problems; 0.001 suits CAD or surveying data.

The reason line is the part a teacher or grader is most likely to ask about, so the calculator shows it on the same line as the verdict.

For map and model work, the scale factor k from the verdict is the same linear ratio Scale Conversion Calculator uses to move between a model dimension and the real-world length.

The proportion is exact, but the inputs and the correspondence decision decide how trustworthy the verdict is.

• • The calculator does not check that the three sides you typed actually form a triangle. A real similarity needs positive sides that close, which is a separate step from the AA, SSS, or SAS test.
• • The proportion a2 / a1 = b2 / b1 = c2 / c1 is silent about which side of the second triangle is which. Two triangles with the same side lengths can be similar and mirror-reflected, which the proportion treats as identical.

If the verdict reads Not similar but you expected Similar, check the pairing of a with a, b with b, and c with c, and check the unit on all six sides.

According to Khan Academy, the AA similarity criterion states that two triangles are similar if two pairs of corresponding angles are equal, which is the foundation for the proportion a2 / a1 = b2 / b1 = c2 / c1.

According to Math Open Reference, two pairs of sides in the same proportion and the included angle equal is the SAS criterion for similarity between triangles.

After the verdict and k are known, the rest of a triangle's properties still need a separate solver, and Triangle Calculator fills in the angles, perimeter, and area from the three sides.