About

Misidentifying a triangle's angles from its side lengths is a common source of error in surveying, structural engineering, and CNC machining. The Law of Cosines generalizes the Pythagorean theorem to non-right triangles: given three sides a, b, and c, each angle is recovered via γ = arccos(a² + b² − c²2ab). This calculator applies that identity to all three vertices, then derives area via Heron's formula, altitudes, medians, circumradius R, and inradius r. Results assume Euclidean flat-plane geometry. For geodetic triangles on Earth's surface exceeding roughly 50 km per side, spherical excess introduces measurable deviation.

Formulas

The Law of Cosines relates each angle to the three side lengths. For a triangle with sides a, b, c opposite to angles α, β, γ respectively:

γ = arccos(a² + b² − c²2ab)

The area is computed via Heron's formula using the semi-perimeter s = a + b + c2:

A = √s(s − a)(s − b)(s − c)

Heights (altitudes) from the area relation:

h_a = 2Aa

Medians via the median length formula:

m_a = 12√2b² + 2c² − a²

Circumscribed and inscribed circle radii:

R = abc4A

r = As

Where: a, b, c = side lengths. α, β, γ = angles opposite the respective sides. s = semi-perimeter. A = area. R = circumradius. r = inradius. h_x = altitude to side x. m_x = median to side x.

Reference Data

Property	Formula	Unit	Notes
Angle α	arccos((b² + c² − a²) ÷ 2bc)	°	Opposite side a
Angle β	arccos((a² + c² − b²) ÷ 2ac)	°	Opposite side b
Angle γ	arccos((a² + b² − c²) ÷ 2ab)	°	Opposite side c
Semi-perimeter s	(a + b + c) ÷ 2	units	Half of perimeter
Area A	√s(s−a)(s−b)(s−c)	units²	Heron's formula
Perimeter P	a + b + c	units	Sum of sides
Height h_a	2A ÷ a	units	Altitude to side a
Height h_b	2A ÷ b	units	Altitude to side b
Height h_c	2A ÷ c	units	Altitude to side c
Median m_a	12√2b² + 2c² − a²	units	To midpoint of a
Median m_b	12√2a² + 2c² − b²	units	To midpoint of b
Median m_c	12√2a² + 2b² − c²	units	To midpoint of c
Circumradius R	abc4A	units	Circumscribed circle radius
Inradius r	As	units	Inscribed circle radius
Triangle inequality	a + b > c	-	Must hold for all permutations
Angle sum	α + β + γ = 180°	°	Euclidean constraint
Right triangle test	a² + b² = c²	-	Pythagorean special case
Equilateral area	√34a²	units²	When a = b = c
Isosceles test	Two sides equal	-	Two angles also equal
Scalene definition	All sides different	-	All angles different
Obtuse test	c² > a² + b²	-	Largest angle > 90°

Frequently Asked Questions

The triangle inequality theorem requires that the sum of any two sides must strictly exceed the third side. If a + b ≤ c for any permutation, no triangle exists in Euclidean geometry. The calculator validates all three inequalities before computing. A degenerate case where equality holds (e.g., sides 3, 4, 7) produces a zero-area collinear segment, not a triangle.

It does not handle them differently. The Pythagorean theorem is a special case of the Law of Cosines. When γ = 90°, cos(90°) = 0, so the cross term 2abcos(γ) vanishes, leaving c² = a² + b². The calculator automatically classifies such triangles as right triangles when the largest angle falls within ±0.001° of 90°.

When two sides nearly sum to the third, the argument of arccos approaches −1 or +1. Floating-point rounding in IEEE 754 double precision can push it outside the valid [−1, +1] domain, producing NaN. This calculator clamps the cosine argument to that domain before applying arccos, preventing computation failure. However, the resulting angle values for near-degenerate triangles carry reduced significant digits. For critical surveying work, verify with independent measurements.

No. This tool assumes Euclidean (flat-plane) geometry where the interior angles sum to exactly 180°. On a sphere, the angle sum exceeds 180° by an amount called spherical excess, proportional to the triangle's area relative to the sphere's radius squared. For Earth-surface triangles with sides under approximately 10 km, the Euclidean approximation error is typically below 0.01°.

The extended Law of Sines states asin(α) = 2R. For obtuse triangles, the circumcenter lies outside the triangle. As the largest angle approaches 180°, the circumradius grows toward infinity because sin(α) approaches zero for the near-straight angle. For equilateral triangles, R = a√3, the minimum circumradius for a given side length.

Classification operates on two independent axes. By sides: equilateral (all three sides equal within tolerance 0.0001), isosceles (exactly two sides equal), or scalene (all different). By angles: acute (all angles < 90°), right (one angle = 90° within 0.001°), or obtuse (one angle > 90°). Both classifications are reported simultaneously, e.g., "Scalene Obtuse".