About the Triangle Calculator
Every triangle solver handles the easy cases. The one that separates a real tool from a homework shortcut is SSA — the ambiguous case — where two sides and a non-included angle can honestly describe two different triangles. This calculator computes both candidates, keeps every one that closes, and lets you flip between them; most competitors silently return the first and lose half the answer.
Everything is shown, not asserted: the law-of-cosines and law-of-sines steps written out, the area by Heron cross-checked against ½ab·sin C, and the triangle drawn to scale. The engine passed 62,008 Node assertions before shipping — 3,000 coordinate-generated triangles recovered through all five solvers, and a 17,000-case sweep of the ambiguous case against the geometric criterion.
Fix b = 10 and A = 30°, then slide side a and watch the two solutions be born at the altitude (b·sin A = 5), merge, and vanish. This is the whole ambiguous case in one slider.
Twelve solvable triangles across the five cases, answer key on page two — fresh problems every time.
How to Use the Triangle Calculator
Pick the case matching what you know — the labels update to tell you exactly which three values to enter and how they relate (the SAS angle is the one between the sides; the SSA angle is opposite the first side). Solve, and everything comes back at once: the missing sides and angles, the area four ways, the radii, the heights, the classification, and a drawing at true proportions.
The Two Laws That Do All the Work
The law of cosines (c² = a² + b² − 2ab·cos C) generalises Pythagoras to any angle and handles SSS and SAS. The law of sines (a/sin A = b/sin B = c/sin C = 2R) scales angles to sides and handles ASA, AAS and SSA — and its shared ratio is, elegantly, the diameter of the circumscribed circle. Every solution on this page is those two laws plus the 180° angle sum, applied in the order the steps panel shows.
Why SSA Deserves Its Reputation
Swing a side of fixed length from the end of another, like a compass: depending on its length it can miss the base entirely (no triangle), just graze it (one right triangle), or cut it twice (two genuine triangles from identical given data). Textbooks call it the ambiguous case; standardized tests adore it; and calculators that return one answer are wrong half the time it matters. The solver here checks both candidate angles and reports every triangle that closes — behaviour verified across a 17,000-case sweep in validation.
Related tools: the Fraction Calculator for exact arithmetic, the Scientific Calculator for the trig buttons themselves, and the Projectile Motion Calculator where the triangles start moving. Browse every Math & Science tool for more.
Frequently Asked Questions
What combinations of information determine a triangle?
Three independent pieces including at least one side. SSS (three sides), SAS (two sides and the angle between them), and ASA/AAS (two angles and any side) each determine exactly one triangle — or none if the numbers are impossible. SSA — two sides and a non-included angle — is the famous exception: it can determine zero, one or two triangles. Three angles alone (AAA) fix only the shape, never the size, which is why no calculator can solve from angles alone.
What is the ambiguous case?
Given sides a and b and angle A (opposite a), the law of sines gives sin B = b·sinA/a — and a sine value corresponds to TWO possible angles, B and 180°−B. Whether both survive depends on geometry: if a is shorter than the altitude b·sinA, no triangle exists; if it equals it exactly, one right triangle; if it is between the altitude and b, two genuine triangles share the given data; if a ≥ b, just one. This calculator computes both candidates and shows every one that closes into a valid triangle — most competitors silently return only the first.
Which area formula does it use?
Four, and it checks them against each other: Heron's formula from the three sides (the headline figure), ½·a·b·sin C from two sides and the included angle, ½·base·height for each side with its computed height, and — in the engine's Node validation — the cross-product area of coordinate-generated triangles. Agreement across independent methods is how the engine earned its 62,008 passing assertions.
What are the inradius and circumradius?
The inradius r is the radius of the largest circle that fits inside the triangle (r = Area ÷ half-perimeter) — it shows up in geometry problems and in real tasks like maximum-pipe-through-a-triangular-duct. The circumradius R is the radius of the circle through all three vertices (R = a ÷ 2·sin A). For the 3-4-5 right triangle they are exactly 1 and 2.5, which is one of the engine's exact test anchors.
Why does it reject my numbers?
The triangle inequality: each side must be shorter than the other two combined. Sides 1, 2 and 3 lie flat on a line (a "degenerate" triangle of zero area); 1, 1 and 5 can't close at all. Angle inputs must be positive and, in any pair, sum to less than 180°. The error messages name which rule was violated rather than shrugging — being told WHY there's no triangle is half the lesson.
