About

Incorrect triangle dimensions propagate errors through every downstream calculation - structural loads, land surveys, CNC toolpaths, and navigation bearings all depend on precise trigonometric resolution. This calculator solves an arbitrary triangle from any valid combination of three known elements (sides a, b, c and angles A, B, C) using the Law of Cosines and the Law of Sines. It outputs all six primary elements plus derived properties: area via Heron's formula, altitudes, medians, angle bisector lengths, inradius r, circumradius R, and the coordinates of centroid, incenter, circumcenter, and orthocenter.

The tool enforces the triangle inequality (a + b > c) and validates that angles sum to 180° before solving. Ambiguous SSA cases (the "ambiguous case") are detected and both solutions are presented when they exist. Note: floating-point arithmetic limits precision to approximately 12 significant digits. For geodetic work on curved surfaces, a planar triangle solver introduces measurable error beyond baselines of roughly 10 km.

Formulas

The solver selects its algorithm based on the input combination. For three sides (SSS), it applies the Law of Cosines in angle-finding form. For combinations involving angles, it uses the Law of Sines and angle-sum property.

Law of Cosines (solving for angle):

C = arccos ( a² + b² − c²2ab )

Law of Cosines (solving for side):

c = √a² + b² − 2ab cos C

Law of Sines:

asin A = bsin B = csin C

Heron's formula for area:

S = √s(s − a)(s − b)(s − c)

Where a, b, c are sides opposite to angles A, B, C respectively. s = a + b + c2 is the semi-perimeter. S is the area. r is the inradius. R is the circumradius. h_a, m_a, t_a are the altitude, median, and angle bisector to side a.

Reference Data

Triangle Type	Condition	Angle Range	Special Properties
Equilateral	a = b = c	All 60°	All centers coincide; maximum area for given perimeter
Isosceles	Two sides equal	Two angles equal	Axis of symmetry through apex; altitude = median = bisector on base
Scalene	All sides different	All angles different	No symmetry; all centers distinct
Right	c² = a² + b²	One angle = 90°	Circumcenter on hypotenuse midpoint; R = c/2
Obtuse	One angle > 90°	90° < max < 180°	Circumcenter outside triangle; orthocenter outside triangle
Acute	All angles < 90°	0° < each < 90°	All centers inside triangle
Degenerate	a + b = c	One angle = 180°	Zero area; collinear points; not a valid triangle
Key Formulas & Constants
Heron's Area	S = √s(s − a)(s − b)(s − c) where s = (a + b + c)/2
Law of Cosines	c² = a² + b² − 2ab cos(C)
Law of Sines	asin A = bsin B = csin C = 2R
Inradius	r = Ss
Circumradius	R = abc4S
Height to side a	h_a = 2Sa
Median to side a	m_a = 12√2b² + 2c² − a²
Bisector to side a	t_a = 2bc cos(A/2)b + c
Euler's Relation	OI² = R(R − 2r) where OI is distance between circumcenter and incenter
Max area (fixed perimeter)	Equilateral: S_max = √336 P²

Frequently Asked Questions

When you provide two sides and an angle opposite one of them (SSA), the Law of Sines may yield two valid values for the unknown angle - one acute and one obtuse. The calculator detects this by checking whether the supplement angle (180° − B) also forms a valid triangle (i.e., the remaining angle stays positive). When two solutions exist, both are displayed with all derived properties computed independently for each.

Three angles alone (AAA) define the shape but not the size of a triangle. Infinitely many similar triangles share the same angle set. Without at least one side length, no unique triangle exists, so the area, perimeter, and all metric properties are indeterminate. You must supply at least one side to anchor the solution.

The calculator places vertex A at the coordinate origin and vertex B on the positive x-axis. Vertex C is computed from side lengths and the included angle using cosine and sine. All notable points are then derived algebraically from these exact coordinates. Precision is limited to IEEE 754 double-precision floating point - approximately 15 significant decimal digits. Results are displayed rounded to 4 decimal places.

No. This is a pure Euclidean geometry solver operating in a flat, ideal plane. It does not account for thermal expansion of physical materials, geodetic curvature of the Earth's surface, or relativistic effects. For physical triangles at scales above roughly 10 km, spherical trigonometry should be used. For structural engineering, apply material-specific tolerances after obtaining the geometric solution.

Euler's inequality states R ≥ 2r for any valid triangle, with equality holding only for equilateral triangles. The inradius r = S/s (area divided by semi-perimeter) represents the largest circle fitting inside the triangle. The circumradius R = abc/(4S) represents the smallest circle enclosing all vertices. The ratio R/r is a measure of how far a triangle deviates from equilateral.

Yes. The calculator identifies the triangle type after solving. If any computed angle equals 90° (within floating-point tolerance of 0.0001°), it labels the triangle as "Right." It also classifies as Acute (all angles below 90°), Obtuse (one angle above 90°), Equilateral, or Isosceles. The Pythagorean theorem is a special case of the Law of Cosines where cos(90°) = 0.