Oblique Triangle Calculator
Oblique triangle solver for SAS, SSS, and ASA with angle sum check.
What it does
Non-right triangles solve from mixed knowns: two sides with their angle, three sides, or two angles with a side. Each case owns one law, cosines for SAS and SSS, sines for ASA.
The tool above takes the mode with its knowns. It returns the missing pieces with the 180 degree check.
The default SAS 10 and 12 around 30 degrees gives b = 6 with A = 56.3 and C = 93.7. SSS 3, 4, 5 returns the 36.9, 53.1, 90 set.
Laws follow OpenStax precalculus with Texas A and M open texts behind them. All three state the same sine ratios and cosine forms used here.
How to use
- Pick SAS, SSS, or ASA mode.
- Enter the known sides or angles.
- Read the main answer first.
- Read the second answer beside it.
- Check the 180 sum line.
- Switch modes to cross-check.
Formula
- a, b, c
- Triangle sides opposite A, B, C.
- A, B, C
- Triangle angles in degrees.
- B / ASA pair
- Included or known angle input.
Angles come from arccosine for stability, never from a lone arcsine. Degree mode throughout; the check line always sums to 180. SSS validates the triangle inequality first.
Worked example
Example: SAS 10, 12, 30 deg
Defaults: SAS mode, a = 10, c = 12, B = 30 degrees.
- SAS: sides 10 and 12 with included 30 degrees.
- b squared = 100 + 144 - 240 x cos30 = 36.154.
- b = root(36.154) = 6.
- cosA = (36.154 + 144 - 100) / (2 x 6 x 12) = 0.5554.
- A = arccos = 56.3 degrees.
- C = 180 - 56.3 - 30 = 93.7 degrees.
- Tool shows b = 6, A = 56.3 deg, C = 93.7 deg.
Reference table
| Knowns | Law | Gives |
|---|---|---|
| SAS | Cosines | Third side, then angles |
| SSS | Cosines | All three angles |
| ASA | Sines | Other two sides |
| SSA | Ambiguous | Not covered here |
Limits
- No SSA ambiguous mode.
- Degrees only, no radians input.
- Huge sides lose fine precision.
- No area-first solving.
Common mistakes
- Running sines on SAS directly.
- Rounding mid solution.
- Feeding SSA into SAS.
- Ignoring inequality failures.
- Mixing radian mode calculators.
Try it yourself
Quick checks. Answers sit tucked under each question.
SSS sides 3, 4, 5?
Arccos set: A = 36.9, B = 53.1, C = 90 degrees.
ASA A = 50, B = 60, c = 10?
C = 70, a = 8.2, b = 9.2 by sines.
SSS sides 1, 2, 10?
1 + 2 fails past 10: violates inequality, no triangle.
Questions
Which law does each mode use?
SAS and SSS run law of cosines; ASA runs law of sines. Pick the mode matching your knowns.
Is there an ambiguous case?
Only SSA, which this page skips. SAS, SSS, and ASA each solve uniquely.
Do 1, 2, 10 make a triangle?
No. Sides 1, 2, 10 violate the inequality, so no triangle exists.
How do I sanity check angles?
Yes: largest side always faces the largest angle, a free validity check on every answer.
When should I round?
Carry full precision to the end, then round sides to 0.1 and angles to 0.1 degrees.
Why cosines instead of sines?
No. Cosines stay stable near 0 and 180 where sines turn ambiguous.