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Triangle Angle Calculator

Enter the three sides of a triangle (SSS) and get all three interior angles by the law of cosines, plus whether the triangle is acute, right or obtuse.

Side a

Side b

Side c

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How it works

cos A = (b² + c² − a²) ÷ 2bc · cos B = (a² + c² − b²) ÷ 2ac · cos C = (a² + b² − c²) ÷ 2ab · A + B + C = 180°

When you know all three sides of a triangle (the SSS case), the law of cosines rearranges to give each angle directly: the cosine of an angle equals the sum of the squares of its two adjacent sides minus the square of the opposite side, over twice the product of those two sides. Arccosine returns a unique value between 0° and 180°, so there is no ambiguity. The largest angle always sits opposite the longest side, and comparing it to 90° tells you whether the triangle is acute, right or obtuse.

Worked example

Take the classic 3-4-5 triangle. Angle C opposite the longest side 5 has cos C = (3² + 4² − 5²) ÷ (2·3·4) = (9 + 16 − 25) ÷ 24 = 0, so C = 90°. Angle B opposite 4: cos B = (9 + 25 − 16) ÷ 30 = 0.6, so B ≈ 53.13°. Angle A opposite 3: cos A = (16 + 25 − 9) ÷ 40 = 0.8, so A ≈ 36.87°. They sum to 180°, and because the biggest angle is exactly 90° it is a right triangle.

Sources: Law of cosines — Wikipedia

Frequently asked questions

What is the triangle inequality?

Each side of a valid triangle must be shorter than the sum of the other two. If one side is as long as or longer than the other two combined, the sides can’t close into a triangle, so no set of angles exists and the calculator flags it.

Why use the law of cosines instead of the law of sines here?

With only the three sides known (no angle–side pair), the law of sines can’t start. The law of cosines works straight from three sides, and arccosine gives a unique angle in 0°–180°, avoiding the acute-versus-obtuse ambiguity that sine has.

How do I know if the triangle is acute, right or obtuse?

Look at the largest angle, which lies opposite the longest side. If it is exactly 90° the triangle is right, more than 90° obtuse, and less than 90° acute. The other two angles are always acute.

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This calculator is for educational purposes. Double-check important results before acting on them.