Triangle Calculator
Solve any triangle from SSS, SAS, AAS, SSA or right-triangle inputs — with a true-to-scale diagram, area, perimeter, inradius, circumradius and the full ambiguous-case check for SSA.
⚡ Quick Tips for Solving Triangles Correctly
Features
SSS, SAS, AAS, SSA and right triangle — covering every common combination of known values.
Automatically detects zero, one or two valid triangles and shows both when genuinely ambiguous.
Drawn from your triangle's actual computed vertices — not a fixed generic placeholder shape.
Both circle radii included by default, alongside area, perimeter and all six sides and angles.
Every result shows the exact Law of Cosines, Law of Sines or Heron's formula step that produced it.
Toggle between 2 and 4 decimal places, and show or hide steps and the diagram independently.
A Diagram That Actually Matches Your Triangle
Most free triangle solvers draw the same static triangle icon no matter what you enter — three sides that form a razor-thin sliver get illustrated identically to a near-equilateral shape. This tool computes real vertex coordinates from whichever sides and angles you solved for, using the same Law of Cosines placement a CAD program would use, then scales the result to fit the display while preserving true proportions. The payoff is genuinely useful, not just decorative: entering a wildly obtuse or near-degenerate triangle by mistake shows up immediately as a visibly stretched or nearly-flat shape, long before you'd notice the same error buried in a table of angle values.
The SSA Case Almost Every Calculator Gets Wrong
Four of the five standard triangle configurations — SSS, SAS, AAS, and the right-triangle case — always produce exactly one triangle, or none at all if the values are impossible. SSA is different: because the Law of Sines can return two valid angles (an acute one and its supplementary obtuse counterpart) for the same sine value, the same two sides and one angle can legitimately describe two entirely different triangles. A calculator that silently picks just one of them is quietly giving an incomplete answer. This tool checks the geometry explicitly — comparing the given side against the height needed to reach the base — and when both solutions are genuinely valid, shows both with a direct toggle between them, rather than presenting a single answer as if no ambiguity existed.
Why Two Circles Come Free With Every Triangle
Every triangle has exactly one circle that fits snugly inside it, touching all three sides (the incircle, radius r = Area/s), and exactly one circle that passes through all three of its corners (the circumcircle, radius R = abc/4×Area). Both come directly from numbers this calculator already computes for every other result — area and the sides — so there's no reason a triangle calculator should force you to look them up in a separate tool. The circumradius specifically matters whenever a real-world problem needs a single circle through three known points — laying out a circular arch from three support points, or triangulating a position from three reference bearings.
The inradius shows up in a quieter but equally practical set of problems: fitting the largest possible circular pipe, tank or drain cover into a triangular gap, or determining how large a circular seal can sit flush against three angled walls. Because r and R are always computed from the same area and side-length numbers already on screen, there's no extra input required and no separate formula to look up — they simply appear as two more rows in the results the moment a valid triangle is solved, on every single mode from SSS through to the right-triangle case.
Who Actually Reaches for a Triangle Calculator
Trigonometry and geometry students checking homework across all four standard configurations, not just the SSS case most textbook chapters lead with. Surveyors and engineers solving field triangulation problems where SSA setups (a known baseline, a measured angle, and a partially-known distance) come up constantly, and the ambiguous case is a real practical concern, not just an academic curiosity. Carpenters and builders checking rafter angles, roof pitches, and truss dimensions, where the diagram doubles as a quick visual confirmation that a cut angle looks right before committing material. Computer graphics and game developers working with triangle-mesh geometry, where every 3D surface reduces to triangle math at the lowest level. For right-triangle-only problems specifically, the Pythagorean Theorem Calculator offers a more focused single-purpose interface.
Two Calculations Worth Walking Through
SSA ambiguity in practice: with side a = 8, side b = 10, and angle A = 40°, the Law of Sines gives sin(B) ≈ 0.803, which corresponds to two valid angles for B — roughly 53.4° and its supplement, 126.6°. Both keep angle sum under 180° when combined with A = 40°, so both are genuinely valid triangles with completely different shapes and areas. Toggling between the two solutions in this calculator shows exactly how differently the same three input numbers can resolve.
Roof pitch check: a carpenter knows the horizontal run (12 ft) and the rafter length (13 ft) of a roof section and needs the rise. That's a leg and the hypotenuse, not two legs — switching Right Triangle mode to "Leg + Hypotenuse" and entering a = 12, c = 13 solves it directly as b = √(13² − 12²) = 5 ft of rise, with the actual roof pitch angle (opposite that rise) working out to 90° − asin(12/13) ≈ 22.6° from horizontal. Recognizing which two values you actually have, out of all three sides, is the first and most important step here — before any arithmetic happens at all.
Why Browser-Based Beats a Physical Protractor and Table of Sines
A printed table of sines and cosines gets you the numbers; it doesn't draw the shape, check the triangle inequality, or catch an SSA ambiguity automatically. This tool runs every calculation and renders the diagram entirely client-side in JavaScript, with nothing you enter ever sent to a server — safe for proprietary engineering figures or homework you'd rather keep private. For trigonometric function evaluation beyond a single triangle — sin, cos, tan at arbitrary angles — the Scientific Calculator handles that directly.
Where This Calculator Draws the Line
It solves single, standalone triangles from the five input combinations covered here — it doesn't handle spherical triangles (used in astronomy and geodesy), 3D tetrahedral geometry, or systems of multiple connected triangles like a full truss structure. For polygon area or perimeter beyond three sides, or circle-specific geometry like arc length and sector area, a dedicated tool for that shape will serve better than trying to decompose it into triangles by hand first.