Solve any oblique (non-right) triangle. Input any 3 known values — sides or angles — and compute all 6 elements, area, perimeter, altitudes, medians, and angle bisectors.
| Element | To side a | To side b | To side c |
|---|---|---|---|
| Altitude | 7.67 cm | 6.71 cm | 5.96 cm |
| Median | 7.76 cm | 7.00 cm | 6.02 cm |
| Angle Bisector | 7.73 cm | 6.87 cm | 5.99 cm |
| Case | Given | Method | Solutions |
|---|---|---|---|
| SSS | 3 sides | Law of cosines for angles | 0 or 1 |
| SAS | 2 sides + included angle | Law of cosines for 3rd side | 1 |
| ASA | 2 angles + included side | 3rd angle = 180°−A−B, law of sines | 1 |
| AAS | 2 angles + opposite side | 3rd angle = 180°−A−B, law of sines | 1 |
| SSA | 2 sides + non-included angle | Law of sines (ambiguous) | 0, 1, or 2 |
An oblique triangle is any triangle that does not contain a right angle. Unlike right triangles, where the Pythagorean theorem and basic trigonometric ratios suffice, oblique triangles require the law of sines and the law of cosines — the two fundamental tools of general triangle solving.
This calculator handles all five standard cases for specifying a triangle: SSS (three sides), SAS (two sides and the included angle), ASA (two angles and the included side), AAS (two angles and a non-included side), and SSA (two sides and a non-included angle, the ambiguous case). For each case, it selects the appropriate solving strategy, computes all six elements (three sides and three angles), and then derives area, perimeter, semi-perimeter, circumradius, inradius, all three altitudes, all three medians, and all three angle bisector lengths.
The law of cosines generalizes the Pythagorean theorem: c² = a² + b² − 2ab·cos(C). It is used whenever at least two sides are known. The law of sines states a/sin(A) = b/sin(B) = c/sin(C) = 2R, connecting sides to their opposite angles and to the circumradius. Together, these two laws can solve any triangle given three independent measurements.
Preset examples cover each case type so you can see how the solver handles different configurations. The SSA case may produce two valid triangles (the "ambiguous case"), and both are displayed when they exist.
Oblique Triangle Solver — All Cases (SSS, SAS, ASA, AAS, SSA) problems often require several dependent steps, and a small arithmetic slip can propagate through every derived value. This calculator is tailored to that workflow: you enter side a, side b, side c, and it returns side a, side b, side c, angle a in one consistent pass. It is useful for homework checks, worksheet generation, tutoring walkthroughs, and fast field/design estimates where you need reliable geometry results without rebuilding the full derivation each time.
Law of cosines: c² = a² + b² − 2ab·cos(C)
Law of sines: a/sin(A) = b/sin(B) = c/sin(C)
Area = √[s(s−a)(s−b)(s−c)] (Heron)
Altitude hₐ = 2·Area/a
Median mₐ = ½√(2b²+2c²−a²)
Angle bisector tₐ = (2bc/(b+c))·cos(A/2)
Circumradius R = abc/(4·Area)
Inradius r = Area/sResult: c ≈ 9.17, A ≈ 73.22°, B ≈ 46.78°, Area ≈ 34.64
c = √(100+64−160·cos60°) = √(84) ≈ 9.17. Angle A = arccos((64+84.07−100)/(2·8·9.17)) ≈ 73.22°. B = 180−73.22−60 = 46.78°. Area = ½·10·8·sin60° ≈ 34.64.
This oblique triangle solver — all cases (sss, sas, asa, aas, ssa) tool links the entered values (side a, side b, side c, angle a (°)) to the target geometry relationships used in class and practice problems. Instead of solving each intermediate step manually, you can validate setup and arithmetic quickly while still tracing which measurements drive the final result.
Formula focus: the calculator formula
Oblique Triangle Solver — All Cases (SSS, SAS, ASA, AAS, SSA) shows up in school geometry, technical drafting, construction layout checks, and early engineering design estimates. When values are changed repeatedly, the calculator helps you compare scenarios quickly and see how sensitive the shape is to each dimension.
Start with the primary outputs (side a, side b, side c, angle a) and then use the remaining cards/tables to confirm consistency with your diagram. Keep units consistent across inputs, and round only at the end if your assignment or project specifies a fixed precision.
Last updated:
An oblique triangle is any triangle with no 90° angle. Both acute triangles (all angles < 90°) and obtuse triangles (one angle > 90°) are oblique.
Three independent measurements (sides or angles), with at least one being a side. Three angles alone only determine the shape (similarity class), not the size.
c² = a² + b² − 2ab·cos(C). It relates the three sides of a triangle to one of its angles and generalizes the Pythagorean theorem.
a/sin(A) = b/sin(B) = c/sin(C) = 2R. It connects each side to the sine of its opposite angle and to the circumradius R.
Because the angle is not between the two known sides, the law of sines gives sin(B) which has two possible angles (B and 180°−B). Both may produce valid triangles.
A median connects a vertex to the midpoint of the opposite side. An altitude is perpendicular from a vertex to the opposite side. An angle bisector divides an angle into two equal halves.
Calculate the area, perimeter, all angles, altitudes, circumradius, and inradius of a triangle from two sides and the included angle (SAS method). Includes triangle classification, presets, and ref...
Solve triangles from two sides and a non-included angle (SSA). Detects 0, 1, or 2 solutions. Computes area, perimeter, all angles/sides, circumradius, inradius, and altitudes for each valid triangle.
Calculate triangle area from three sides using Heron's formula. Also computes perimeter, all angles, altitudes, medians, circumradius, inradius, and classifies by angle and side type.