Triangle Solver
Solve triangles from any combination of 3 sides + angles (SSS, SAS, ASA, AAS, SSA). Outputs all 3 sides, all 3 angles, area, perimeter, triangle type. Uses Law of Cosines + Sines + Heron's formula.
Triangle Solver
How to use the Triangle Solver
Identify what you know
Enter exactly 3 known values: any combination of sides + angles, with at least one side. The standard configurations: SSS (3 sides), SAS (2 sides + included angle), ASA (2 angles + included side), AAS (2 angles + non-included side), SSA (2 sides + non-included angle — ambiguous case).
Match labels correctly
Side a is opposite angle A, side b opposite B, side c opposite C. This is the universal convention. If you\'re working from a diagram with different labels, just relabel — the math doesn\'t care which letter you use, as long as the side-angle pairing is correct.
Read all properties
The tool returns: all 3 sides, all 3 angles, area (via Heron\'s formula), perimeter, and triangle type (acute/right/obtuse + equilateral/isosceles/scalene). The "Configuration" badge tells you which case (SSS/SAS/etc.) was solved.
Watch for the SSA ambiguous case
SSA configurations can have 0, 1, or 2 valid triangles. This tool returns the acute-angle solution; an obtuse second solution may also exist. If your problem context requires the obtuse case, compute the supplement (180° − returned angle) for the unknown angle, then recompute the other unknowns.
Solving any triangle with Law of Cosines + Law of Sines
Triangle solving is one of trigonometry\'s most-applied skills. Surveyors triangulate property boundaries; navigators compute distances at sea; engineers analyse truss forces; carpenters cut rafters at specific angles. Five canonical configurations (SSS, SAS, ASA, AAS, SSA) cover all valid input combinations of sides + angles, and two fundamental laws — the Law of Cosines and the Law of Sines — handle every case. This tool implements both, automatically detects which configuration you\'ve given, applies the right approach, and returns all unknowns.
Law of Cosines + Law of Sines — when to use each
Law of Cosines: c² = a² + b² − 2ab·cos(C). Generalises Pythagoras to non-right triangles. Use for: SSS (find any angle from 3 sides) + SAS (find third side from 2 sides + included angle). At C = 90°, cos(C) = 0 and the formula reduces to c² = a² + b² (Pythagoras). Law of Sines: a/sin(A) = b/sin(B) = c/sin(C). Says the ratio of side to opposite-angle sine is constant within a triangle. Use for: ASA, AAS, SSA — anytime you know an angle + its opposite side, you can find any other side from its opposite angle (or vice versa). Heron\'s Formula (area): A = √(s(s−a)(s−b)(s−c)), where s = (a+b+c)/2. Computes area from sides alone — no angles needed.
Law of Cosines for SSS + SAS. Law of Sines for ASA / AAS / SSA. Together they solve any valid triangle configuration in 2-3 algebra steps.
The SSA ambiguous case
SSA (two sides + non-included angle) is the tricky configuration because it can produce 0, 1, or 2 valid triangles. Given side a, side b, and angle A (with a opposite to A): (1) compute h = b·sin(A) — the altitude from the vertex of angle A to side a; (2) if a < h: no valid triangle (the side a is too short to reach side b\'s endpoint); (3) if a = h: exactly one right triangle; (4) if h < a < b: two triangles possible — one acute, one obtuse; (5) if a ≥ b: exactly one triangle. This tool returns the acute-angle case by default. For the obtuse case, the second valid angle B\' = 180° − B. In real applications: SSA usually arises in surveying + navigation; context typically dictates which solution is correct (a far-away mountain vs a nearby hill).
Common triangle-solving applications
Surveying + land measurement: properties divided into triangular plots; areas computed via Heron\'s formula or trigonometry. Navigation: ship\'s position triangulated from 2 known landmarks + bearings (ASA case). Construction: rafter angles for roof pitches; truss design with diagonal braces; bridge engineering. Astronomy: stellar parallax — Earth\'s orbit baseline + observed angles to a star → distance. Computer graphics: 3D rendering uses triangles as the fundamental polygon; rotation + scaling involve repeated triangle-trigonometry. Aviation: dead reckoning + position fixes; triangular flight paths for waypoint navigation. Sports: throwing/kicking trajectories often modelled as parabolic + triangular sub-problems.
10 Things to Know About Triangles
Triangle angles sum to 180° always (in Euclidean geometry). On a sphere, they sum to more — basis of spherical trigonometry.
5 valid input configurations: SSS, SAS, ASA, AAS, SSA. AAA alone has infinite similar triangles (no fixed scale).
Law of Cosines: c² = a² + b² − 2ab·cos(C). Generalises Pythagoras for any angle.
Law of Sines: a/sin(A) = b/sin(B) = c/sin(C). Ratio of side to opposite-angle sine is constant.
Heron\'s formula: area = √(s(s−a)(s−b)(s−c)). Computes area from sides alone, no angles needed.
SSA ambiguous case: can have 0, 1, or 2 valid triangles. Context determines which solution applies.
Triangle types: acute (all angles < 90°), right (one = 90°), obtuse (one > 90°).
Side classification: equilateral (all sides equal), isosceles (two equal), scalene (all different).
Triangle inequality: sum of any 2 sides must exceed the 3rd. Violations indicate impossible triangles.
Triangles are the foundation of 3D computer graphics — every 3D model is ultimately rendered as collections of triangles.
Frequently Asked Questions
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Three common causes. (1) Triangle inequality violation: any single side exceeds the sum of the other two. Example: sides 1, 2, 10 → impossible (1+2 < 10). (2) Angle sum mismatch: if 3 angles sum to anything other than 180°. (3) SSA produces no triangle: the given side is too short to "reach" — see SSA ambiguous case. Check your inputs match the geometric reality.
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Standard convention: vertex A has side a OPPOSITE to it (not adjacent). Same for B/b and C/c. Looking at a triangle: pick any vertex, label it A; the side directly across from it (not touching A) is side a. Why this convention: makes the Law of Sines (a/sin A = b/sin B = c/sin C) clean and memorable. If your diagram uses different labels, rename to match this convention before entering.
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Given two sides + a non-included angle, you can sometimes draw TWO different triangles that satisfy the same constraints — one acute, one obtuse. Geometrically: imagine the known angle\'s "hinge"; the other known side can swing in two positions both touching the third side. The tool returns the acute solution by default. For the obtuse case: angle B\' = 180° − B (the returned one); recompute other unknowns accordingly. Real-world example: navigation, where the "near" vs "far" peak both satisfy your bearing data.
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SAS (Side-Angle-Side): angle is BETWEEN the two known sides. Uniquely determines the triangle. SSA (Side-Side-Angle): angle is NOT between the known sides (opposite one of them). Ambiguous — may give 0, 1, or 2 triangles. Examples: SAS — "two sides 5 and 7 with the 60° angle between them"; SSA — "two sides 5 and 7, with the 60° angle opposite to the 7." Tool detects which case automatically based on input.
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3 angles only fix the SHAPE of the triangle (its proportions), not the SIZE. Infinite similar triangles exist at every possible scale. To solve uniquely, you need at least one side. Conceptually: a 30-60-90 triangle could be 1cm wide or 1 km wide — same angles either way.
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Limited only by floating-point precision (~15-16 significant digits in JavaScript). For practical surveying/construction (3-5 significant digits), results are exact. For ill-conditioned cases (very obtuse or very thin triangles), small input errors get amplified — measure carefully. For high-precision geodesy or astronomy, spherical trigonometry + Earth\'s curvature need explicit handling — this tool is planar (Euclidean) only.
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This tool uses degrees. To convert your radian inputs: multiply by 180/π. Many physics + engineering contexts use radians; conversion is a single multiplication. Examples: π/4 rad = 45°; π/3 rad = 60°; π/2 rad = 90°. For most school + practical work, degrees are more intuitive; for advanced math + physics, radians.
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Right triangles (one 90° angle) are a special case. Pythagorean theorem: a² + b² = c² (where c is hypotenuse opposite right angle). SOH-CAH-TOA: sin = opposite/hypotenuse; cos = adjacent/hypotenuse; tan = opposite/adjacent. Quick approach: enter 90 as one angle in this tool; it handles right triangles correctly via the general laws. Or just use Pythagorean + basic trig directly — simpler for hand calculation.
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No. All calculations run in your browser via JavaScript. Open DevTools → Network and confirm zero outbound requests. Inputs stay on your device. Safe for surveying data, construction plans, or homework.
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Pair with: Quadratic Equation Solver (RT-CNV-094) for algebraic equations; Ohm\'s Law Calculator (RT-CNV-096) for electrical math; Area Calculator (RT-HOM-XXX) for shape areas. External: GeoGebra for visual geometry; Desmos for graphing; Wolfram Alpha for general math; surveying software (Civil 3D, Trimble Business Center) for professional work.
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