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Triangle Angles Calculator

Find a triangle’s angles from its sides.

How it is calculated

Given the three side lengths, the tool uses the law of cosines: A = arccos((b² + c² − a²) / (2bc)) and B = arccos((a² + c² − b²) / (2ac)). The third angle is the rest of the 180° total: C = 180 − A − B. Angle A sits opposite side a, B opposite b and C opposite c. Sides can be in any one unit, because only their ratios matter. All three must be above zero and satisfy the triangle inequality, meaning each side is shorter than the sum of the other two; otherwise there is no result. Angles are shown in degrees with two decimals.

Worked example

Sides a = 3, b = 4, c = 5. A = arccos((16 + 25 − 9) / (2·4·5)) = arccos(32/40) = arccos(0.8) = 36.87°. B = arccos((9 + 25 − 16) / (2·3·5)) = arccos(18/30) = arccos(0.6) = 53.13°. C = 180 − 36.87 − 53.13 = 90°. The angles are 36.87 °, 53.13 ° and 90.00 °, so this is a right triangle.

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Common mistakes

  • Mixing units between sides, such as one in centimetres and another in metres.
  • Entering sides that cannot form a triangle, such as 2, 3 and 6; the longest side must be shorter than the other two combined.
  • Matching an angle to the wrong side. Angle A is opposite side a, not next to it.
  • Entering angles when the tool needs side lengths.
  • Expecting exact whole degrees; the third angle is found by subtraction, so rounding in A and B carries over to C.

About this calculator

Given the three side lengths of a triangle, this calculator finds all three angles using the law of cosines, cos(A) = (b² + c² − a²) / (2bc), applied once per angle and then checked so the three sum to 180°.

It's used by surveyors and carpenters framing angled cuts like roof trusses or staircases where only the side lengths are easy to measure directly, and by students working through trigonometry problems that go beyond the simpler right-triangle cases.

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Comments (1)

  • Editorial team

    The three sides must satisfy the triangle rule, where the two shorter ones add up to more than the longest, or the tool shows no result. A near-miss such as 2, 3 and 5 is flat, not a triangle. A practical check: the largest angle always sits opposite the longest side, and the three angles must total 180 degrees. Handy for roof or frame cuts, but you still need measured sides. Which job are you using it for?

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