Triangle Angle Calculator

Enter a triangle's three side lengths to find all three interior angles using the Law of Cosines, plus an automatic acute/right/obtuse classification.

Quick Facts

Law of Cosines
cos(A) = (b² + c² − a²) / (2bc)
Solves for any interior angle from the three side lengths.
Angle sum
A + B + C = 180°
True for every triangle in Euclidean geometry.
Triangle inequality
a + b > c (and the other two pairs)
Any two sides must outmeasure the third or no triangle exists.

Your Results

Calculated
Angle A
-
Opposite side a
Angle B
-
Opposite side b
Angle C
-
Opposite side c
Triangle Type
-
Based on the largest angle

Ready

Enter three side lengths, then press Calculate.

Formula and Method for Triangle Angle Calculation

Any triangle can be fully solved for its three interior angles if you know the lengths of all three sides (the SSS case). This calculator uses the Law of Cosines — a generalization of the Pythagorean theorem that works for any triangle, not just right triangles — to find each angle directly from the side lengths, then classifies the triangle as acute, right, or obtuse.

How the calculation works

Label the sides a, b, and c, with angle A opposite side a, angle B opposite side b, and angle C opposite side c. The Law of Cosines states c² = a² + b² − 2ab·cos(C). Solving for the angle gives cos(A) = (b² + c² − a²) / (2bc), and the same pattern for B: cos(B) = (a² + c² − b²) / (2ac). Taking the inverse cosine (arccos) of each result gives angles A and B. Because every triangle's interior angles sum to exactly 180°, the third angle follows without another trig calculation: C = 180° − A − B. The calculator then compares the largest angle to 90° to label the triangle acute (all angles under 90°), right (one angle equal to 90°), or obtuse (one angle over 90°).

Common mistakes

  • Invalid side combinations: not every three lengths form a triangle. The triangle inequality requires that the sum of any two sides exceed the third (e.g., sides 2, 3, and 6 cannot form a triangle because 2 + 3 is not greater than 6).
  • Mismatched side-angle pairs: side a must be the side opposite angle A, not adjacent to it — swapping which side is "opposite" which angle changes the Law of Cosines result.
  • Degrees vs. radians: most everyday work uses degrees, but trigonometric functions in code and calculators often default to radians. Confirm which unit you need before applying the result elsewhere.

Real-world applications

  • Surveying and construction use the Law of Cosines to find unknown angles from measured distances, such as staking out a lot corner or checking a roof pitch.
  • Navigation and triangulation problems solve for a bearing or heading angle once three distances are known.
  • Engineering and CAD work verify that a fabricated or drawn triangle matches its intended angles before parts are cut.
  • Physics problems that involve resultant vectors or force triangles often reduce to solving a triangle from three known magnitudes.

Frequently Asked Questions

How do you calculate a triangle's angles from its three side lengths?
Use the Law of Cosines: cos(A) = (b² + c² − a²) / (2bc), then take the inverse cosine (arccos) to solve for angle A. Repeat the same formula to solve for angle B, then find angle C as 180° − A − B.
What is the Law of Cosines?
The Law of Cosines relates a triangle's three sides to one of its angles: c² = a² + b² − 2ab·cos(C). Rearranged to solve for the angle, it becomes cos(C) = (a² + b² − c²) / (2ab), and it works for any triangle, not just right triangles.
Why do a triangle's interior angles always add up to 180°?
This is a basic property of Euclidean (flat) geometry: for every triangle, the three interior angles always sum to exactly 180°, no matter the triangle's shape or size.
What side lengths can actually form a triangle?
Three lengths form a valid triangle only if the triangle inequality holds: the sum of any two sides must be greater than the third side (a + b > c, a + c > b, and b + c > a). If this fails, the sides cannot close into a triangle.

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