Triangle Theorems Calculator: Solve Any Triangle (SSS–AAA)
Solve any triangle from 3 known sides or angles — SSS, SAS, ASA, AAS, SSA, or AAA — and get every side, angle, area, perimeter, and radius.
The six triangle cases
Any triangle is completely determined by 3 of its 6 measures (three sides a, b, c; three angles A, B, C, each opposite the side of the same letter) — as long as those 3 aren't all angles alone. This calculator recognizes which case your 3 known values form and picks the matching theorem:
| Case | Known | Method |
|---|---|---|
| SSS | Three sides | Law of cosines finds every angle |
| SAS | Two sides + the included angle | Law of cosines finds the third side, then the remaining angles |
| ASA | Two angles + the included side | Angle sum finds the third angle, then law of sines finds the sides |
| AAS | Two angles + a non-included side | Same as ASA — the calculation doesn't care which side is given |
| SSA | Two sides + a non-included angle | Law of sines — the ambiguous case: 0, 1, or 2 triangles |
| AAA | Three angles | Angle sum only confirms the shape — no unique size |
Solving SSS — three sides
Example: a = 7, b = 8, c = 9. The law of cosines gives every angle: , then and (summing to 180°). Semi-perimeter , so Heron's formula gives area , inradius , and circumradius .
Solving SAS — two sides + included angle
Example: b = 8, c = 9, angle A = 48.19° (the angle between b and c). The law of cosines solves the third side directly: — the same 7-8-9 triangle as above, worked in reverse.
Solving ASA / AAS — two angles + one side
Example: A = 40°, B = 60°, side a = 10 (opposite A — the AAS case). The third angle is , then the law of sines ratio solves the rest: , .
Solving SSA — the ambiguous case
Example: a = 6, b = 8, angle A = 40° (opposite side a, not between a and b). Solving gives two valid angles, and — both keep under 180°, so both close into genuine, different triangles.
AAA — angles only, no unique size
Example: A = B = C = 60°. The angles sum to 180°, confirming an equilateral SHAPE, but with no side length given there's no way to pin down a size — every scaled copy of that shape (side 1, side 100, side anything) shares these exact angles.
Common mistakes
- Reaching for the law of sines on SSS/SAS. Neither case has a complete side/angle pair to start a sine ratio — use the law of cosines first.
- Missing the second SSA solution. The ambiguous case silently drops a mathematically valid triangle if only one candidate angle is checked.
- Forgetting the angle-sum rule. ASA/AAS/AAA all need minus the known angles to find the last one before anything else can be solved.
Where this shows up
Surveying, navigation, and structural engineering all reduce to "solve a triangle from partial measurements" — this calculator is the general-purpose version. For a deep dive into just one case, see the law of cosines (SSS/SAS) and law of sines (ASA/AAS/SSA) calculators, or the special-case equilateral and right triangle calculators when extra symmetry is already known.
Frequently asked questions
- What is the triangle theorems calculator?
- A general triangle solver that covers all six classic cases — SSS, SAS, ASA, AAS, SSA, and AAA — by combining the angle-sum rule (180°), the law of cosines, and the law of sines. Pick which three of the six measures (three sides, three angles) you know, and it solves the rest.
- Which theorem does each case actually use?
- SSS and SAS use the law of cosines (no side/angle pair is available). ASA, AAS, and SSA use the law of sines once the angle-sum rule fixes the third angle (or, for SSA, once the ambiguous angle is found). AAA only uses the angle-sum rule — there's no side to solve for.
- What's the difference between ASA and AAS?
- Both give two angles plus one side, and both solve identically once you notice knowing two angles means knowing all three (180° − their sum). ASA's side sits BETWEEN the two known angles; AAS's side is opposite one of them. Neither distinction changes the calculation.
- What is the SSA ambiguous case?
- Given two sides and a non-included angle, there can be zero, one, or two valid triangles, because the sine of an angle θ equals the sine of its supplement 180°−θ. This calculator checks both candidate angles and reports every triangle that actually closes to 180°.
- Why does AAA only give angles, not a full triangle?
- Three angles fix a triangle's SHAPE but not its SIZE — infinitely many similar triangles (scaled copies of each other) share the same three angles. You need at least one side length to pin down an actual size, area, or perimeter.
- Why do some inputs give 'no triangle' instead of an answer?
- Two reasons: the triangle inequality (SSS/SAS) — every side must be shorter than the sum of the other two, so sides like 1, 1, 10 can never close — or an impossible angle sum (ASA/AAS/AAA), where the given angles already total 180° or more.
- Does this calculator also find the area, perimeter, and radii?
- Yes — once a triangle is fully solved, the area comes from Heron's formula, the perimeter is the sum of the three sides, the inradius is area ÷ semi-perimeter, and the circumradius is (side·side·side) ÷ (4 × area).
- For a=7, b=8, c=9, what is the area and the inradius/circumradius?
- Semi-perimeter s = 12, so the area (Heron's formula) is √(12·5·4·3) = √720 ≈ 26.83. The inradius r = area/s ≈ 2.24, and the circumradius R = abc/(4·area) ≈ 4.70.