Last updated October 7, 2026 · Report an error
Two triangles are similar if they have exactly the same shape — meaning all their corresponding angles are equal — even if they're different sizes, like a photograph and its enlarged print. Similarity is one of the foundational concepts in geometry, underlying everything from map scales and architectural blueprints to the way shadows and light rays create proportional triangles in basic trigonometry setups. This calculator checks similarity two established ways: SSS (comparing all three side ratios) and SAS (comparing two side ratios along with the angle between them).
How it works
Choose SSS or SAS, then enter the relevant measurements for both triangles. For SSS similarity, the calculator sorts each triangle's three sides from smallest to largest, then compares whether the ratio between corresponding sides is the same across all three pairs — if it is, the triangles are similar. For SAS similarity, the calculator instead compares the ratio of two corresponding sides in each triangle and checks whether the angle between those two sides matches in both triangles — if both the side ratio and the included angle agree, the triangles are similar. In either case, once similarity is confirmed, the common side ratio is reported as the scale factor between the two triangles.
- Enter similarity test.
- Enter triangle 1: side a.
- Enter triangle 1: side b.
- Enter triangle 1: side c (SSS only).
- Enter triangle 1: included angle (SAS only).
- Enter triangle 2: side a.
- Enter triangle 2: side b.
- Enter triangle 2: side c (SSS only).
- Enter triangle 2: included angle (SAS only).
- Click Calculate to see your results.
Examples
Triangles (3,4,5) and (6,8,10)
Every side ratio is exactly 2 (6/3 = 8/4 = 10/5 = 2), so they're similar by SSS with a scale factor of 2 — the second triangle is exactly twice the size of the first.
Triangles that are NOT similar
Triangles with sides (3,4,5) and (6,8,11) fail the SSS test, since 6/3 = 2 and 8/4 = 2 but 11/5 = 2.2 — the third ratio doesn't match the other two, so these triangles are not similar despite two matching ratios.
SAS similarity with an included angle
Two triangles with sides of 4 and 6 (ratio 1.5 relative to a second triangle's sides of 6 and 9) and matching 50° included angles between those sides are similar by SAS, with a scale factor of 1.5.
Who should use it
- Geometry homework verifying triangle similarity.
- Scale-drawing and map-reading problems involving proportional triangles.
- Verifying proportional relationships in architectural or engineering scale models.
Industry applications
- Geometry education
- Architecture and scale-model design
- Surveying (using similar triangles for indirect measurement)
Advantages
- Checks similarity using both established methods (SSS and SAS).
- Reports the scale factor once similarity is confirmed.
- Clear, direct verification without needing to measure every angle.
Limitations
- Doesn't check AA (angle-angle) similarity directly — only the side-based SSS and SAS tests.
Common mistakes to avoid
- Matching up sides in the wrong order — always compare sides after sorting each triangle's sides from smallest to largest.
- Assuming two matching side ratios out of three is enough for SSS similarity — all three must match.
- Confusing similarity (same shape, possibly different size) with congruence (same shape AND size).
Best practices
- Sort each triangle's sides consistently (smallest to largest) before comparing ratios, to ensure you're comparing genuinely corresponding sides.
- For SAS, double-check that the angle you're comparing is actually the included angle between the two sides being compared, not a different angle.
- Use the scale factor to double-check your work — multiplying every side of the smaller triangle by the scale factor should reproduce the larger triangle's sides exactly.
Tips
- If SSS fails but you suspect similarity, double check you've correctly matched each triangle's sides in corresponding order.
- Remember congruent triangles are just similar triangles with a scale factor of exactly 1.