Triangle Similarity Calculator

How to use this triangle similarity calculator

What this calculator does

This calculator tests whether △ABC and △DEF are similar under the SSS, SAS, or ASA criterion, or solves one missing corresponding side when the triangles are assumed to be similar. It also reports a linear scale factor, side-ratio spread, perimeters, areas, and area factor whenever the selected measurements are sufficient. It verifies triangle inequalities and angle sums, so a set of positive numbers that cannot form a real triangle is rejected. The result is a geometric calculation, not a proof that your drawing labels or correspondence are correct; the vertex order A↔D, B↔E, and C↔F must match the problem.

When to use it

• Check a homework or design problem where three corresponding sides are known.
• Confirm SAS similarity from two proportional side pairs and equal included angles.
• Verify ASA similarity from two equal angle pairs and an included corresponding side.
• Recover one missing side in a scale drawing, model, survey sketch, or proportional triangle problem.

How to calculate

1. The calculator opens with a complete 3 – 4 – 5 and 6 – 8 – 10 demonstration. Its results and a validated example XLSX workbook are available immediately.
2. Choose Type. Use Check similarity to test a theorem, or Find the missing side and leave exactly one of the six side boxes blank.
3. For a similarity check, choose Similarity criterion: SSS, SAS, or ASA. Replace the displayed measurements with your own values. Lengths may use any one consistent unit; angles are entered in degrees.
4. Read Similarity result first, then review the scale, spread, perimeter, area, and comparison table. Select Download Excel to export the current canonical inputs and results.
5. Reset clears the demonstration values and all calculated content. Download Excel is then disabled until a complete valid state is entered again.

Input guide

Type is required and accepts one of the two listed workflows. Similarity criterion is required only for a similarity check. AB (a), BC (b), and AC (c) are the three positive side lengths of △ABC; DE (d), EF (e), and DF (f) are their corresponding positive lengths in △DEF. Plain decimals such as 4.5 and standard comma grouping such as 1,250.5 are accepted; decimal-comma notation and scientific notation are rejected to avoid ambiguity. In SSS mode all six sides are required. In missing-side mode exactly one is blank and the remaining corresponding pairs must imply one consistent scale factor. A common error is pairing sides by visual orientation instead of by the stated vertex order.

Angle A and Angle B correspond to Angle D and Angle E. Each required angle must be greater than 0° and less than 180°, and the two entered angles in a triangle must total less than 180°. SAS uses Angle B and Angle E because they are included between AB – BC and DE – EF. ASA uses both angle pairs plus the included sides AB and DE; the third angles are derived from 180° and the remaining sides are reconstructed with the law of sines. For example, 36.869898° and 90° leave a third angle of about 53.130102°. Entering an exterior angle or a radian value as though it were degrees will produce the wrong interpretation.

Output guide

The header pills Mode, Criterion, Status, and Scale provide a compact restatement of the active workflow and result. Similarity result states whether the selected criterion passes, or names the solved side in missing-side mode. Scale factor DEF ÷ ABC is the common linear multiplier k: values above 1 enlarge △ABC, values between 0 and 1 reduce it, and 1 means equal size. Ratio spread is the relative difference among the tested side ratios; zero is exact agreement. ABC perimeter and DEF perimeter are side sums. ABC area and DEF area are computed from the available sides or the SAS included-angle formula. Area factor DEF ÷ ABC equals k² when the triangles are similar. Solved side is populated only in missing-side mode. The table repeats each corresponding pair, perimeter, and area from the same model used by the workbook.

Worked example

The opening example uses AB = 3, BC = 4, AC = 5 and DE = 6, EF = 8, DF = 10. The three DEF-to-ABC ratios are 6÷3 = 2, 8÷4 = 2, and 10÷5 = 2, so SSS similarity passes and k = 2. The perimeters are 12 and 24, which also have ratio 2. The areas are 6 and 24 square units, giving an area factor of 4 = 2². These exact values appear in the first rendered result, comparison table, and exported workbook.

For a broader explanation, OpenStax shows why corresponding sides of similar figures are proportional, while Mathematics LibreTexts summarizes the SSS, SAS, and angle-based similarity tests.