Enter Values

Provide exactly 3 values (at least 1 side). Sides a, b, c are opposite to angles A, B, C respectively.

Sides

Side a

Side b

Side c

Angles (degrees)

Angle A (°)

Angle B (°)

Angle C (°)

Triangle Diagram

Summary

Sides & Angles

Element	Value

Heights, Medians & Bisectors

Element	Value

Formulas & Reference

Law of Cosines

Used to find an unknown side when two sides and the included angle are known (SAS), or to find an unknown angle when all three sides are known (SSS).

c² = a² + b² − 2ab · cos(C)

Law of Sines

Relates the lengths of sides to the sines of opposite angles. Used for ASA, AAS, and SSA cases.

a / sin(A) = b / sin(B) = c / sin(C)

Heron's Formula (Area)

Calculates area from three sides using the semiperimeter s = (a + b + c) / 2.

Area = √(s(s − a)(s − b)(s − c))

Other Area Formula

Area = ½ · a · b · sin(C)

Heights (Altitudes)

The altitude from vertex A to side a is computed from the area.

h_a = 2 · Area / a

Medians

A median connects a vertex to the midpoint of the opposite side.

m_a = ½ · √(2b² + 2c² − a²)

Inradius & Circumradius

r = Area / s R = a / (2 · sin(A))

Angle Bisectors

The length of the bisector from vertex A to side a.

t_a = (2 · b · c · cos(A/2)) / (b + c)

Triangle Types

By sides: Equilateral (all equal), Isosceles (two equal), Scalene (all different)
By angles: Acute (all < 90°), Right (one = 90°), Obtuse (one > 90°)

The Ambiguous SSA Case

When two sides and a non-included angle are known (SSA), the Law of Sines may yield zero, one, or two valid triangles. This happens because sin(B) = b · sin(A) / a can have two solutions (B and 180° − B) if sin(B) < 1. The calculator checks both possibilities: if the supplementary angle also produces a valid triangle (all angles positive and summing to 180°), it reports an ambiguous case and shows the first valid solution. If only one solution is geometrically valid, that solution is shown.

vTriangle