Triangle Angle Calculator

Third angle from two known angles Formula for third angle from two angles Angle A Angle B

Law of Cosines

Given all three sides of a triangle (a, b, c), you can find any angle using the law of cosines:

cos(A) = (b² + c² - a²) / (2bc)

cos(B) = (a² + c² - b²) / (2ac)

cos(C) = (a² + b² - c²) / (2ab)

Take the arccos (inverse cosine) of each result to get the angle.

Triangle Angle Sum

The three interior angles of any triangle always sum to 180°:

A + B + C = 180°

If you know two angles, subtract their sum from 180° to find the third. This is the fastest method when available.

The calculator uses this property to validate results — after computing all three angles via the law of cosines, it checks that the sum is 180° (within rounding tolerance).

Triangle Types by Angles

Acute triangle: All three angles are less than 90°.

Right triangle: One angle equals exactly 90°. The side opposite the right angle is the hypotenuse (longest side).

Obtuse triangle: One angle is greater than 90°. The side opposite the obtuse angle is longest.

Law of Sines Comparison

The law of sines also relates sides and angles: a/sin(A) = b/sin(B) = c/sin(C) = 2R (where R is the circumradius). However, the law of sines is ambiguous for finding angles — arcsin returns an acute angle, but the true angle might be obtuse. The law of cosines avoids this ambiguity, which is why it is the preferred method for finding angles from three sides.

Applications

Surveying: Determine angles of triangular land parcels from measured side lengths using triangulation.

Construction: Verify corners and angles when only side measurements are available.

Navigation: Calculate bearings and course corrections from distance measurements.

Education: Practice using the law of cosines and understanding its advantage over the law of sines for angle determination.