Triangle Angle From Three Sides Calculator
Written by Thierno Sadou Diallo, formula verified per our methodology • Last checked on 9/10/2026
The angle of a triangle opposite a given side is calculated from the three sides with A = acos((b² + c² − a²) ÷ (2bc)), where a is the side opposite the sought angle. For a right triangle with sides 3, 4, and 5, the angle opposite the side of length 5 (the hypotenuse) is exactly 90°.
Explanation
This calculator inverts our law of cosines calculator: where that one gives a side from two sides and the angle between them, this one recovers an angle from the three sides of the triangle, with no angle needing to be known at the start. It is the only direct method to determine the angles of any triangle when only the three side lengths are known — useful in triangulation, surveying, or to check whether a triangle measured in the field is actually right-angled. Two notable special cases allow this formula to be checked completely independently: in a right triangle, the angle opposite the hypotenuse is always exactly 90° (a direct restatement of the Pythagorean theorem, since a² = b² + c² makes the numerator of the formula exactly zero); in an equilateral triangle, the three angles are each exactly 60°, a direct consequence of its perfect symmetry — confirmed by this calculator's second test case.
Example: right triangle with sides 3, 4, and 5
Inputs
Side a (opposite the sought angle): 5 m. Side b: 3 m. Side c: 4 m.
Calculation
A = acos((3² + 4² − 5²) ÷ (2×3×4)) = acos((9+16−25) ÷ 24) = acos(0 ÷ 24) = acos(0) = 90°.
Result
The angle opposite the 5 m side is exactly 90° — this triangle is indeed right-angled.
Frequently asked questions
How do you recover the other two angles of the triangle?
Apply the same formula by simply changing which side plays the role of side "a" (opposite the sought angle): to find the angle opposite side b, use B = acos((a² + c² − b²) ÷ (2ac)), and likewise for the angle opposite c. As a check, the sum of a triangle's three angles is always exactly 180°.
How do you check whether a triangle is right-angled from its three sides?
Calculate the angle opposite the longest side with this calculator: if it is exactly 90°, the triangle is right-angled (this is a direct restatement of the Pythagorean theorem, since the numerator of the formula becomes exactly zero precisely when the square of the longest side equals the sum of the squares of the other two). Any other result indicates a non-right triangle, acute if the angle is less than 90°, obtuse if greater.
What happens if the three sides cannot form a real triangle?
If the three lengths do not satisfy the triangle inequality (each side must be shorter than the sum of the other two), the expression inside the acos() function falls outside the interval [−1, 1], where the cosine is never defined. This calculator then shows an empty result rather than a meaningless number, exactly the same behavior as for a geometrically impossible triangle in our triangle median calculator.