How to Calculate the Angles' Measures of a Triangle, Given the Lengths of the Three Sides

Learn to calculate the measures of the angles of a scalene triangle, given the lengths of its three sides

How to calculate the measures of the angles of the scalene triangle?



To calculate any angle's measure, use one of the three formulas below.

These formulas are directly derived from, and a rearrangement for angles, of the Law of Cosine (or generalized Pythagorean Theorem).


  • m(∡A) = arccos((b2 + c2 - a2) / (2 × b × c))
  • m(∡B) = arccos((a2 + c2 - b2) / (2 × a × c))
  • m(∡C) = arccos((a2 + b2 - c2) / (2 × a × b))
  • Where:
  • a = the side of the triangle opposite angle ∠A
  • b = the side of the triangle opposite angle ∠B
  • c = the side of the triangle opposite angle ∠C


  • Example of angles calculation

  • Calculate the measures of the angles of the triangle knowing the lengths of the three sides: 4, 6, 3.
  • Apply the formulas above.

Calculate the measure of angle ∠A

  • m(∡A) =
  • arccos((62 + 32 - 42) / (2 × 6 × 3)) =
  • arccos((36 + (3 + 4) × (3 - 4))) / 36) =
  • arccos((36 + 7 × - 1)) / 36) =
  • arccos((36 - 7)) / 36) =
  • arccos(29/36) =
  • arccos(0.805555555556) =
  • 0.63418384082329 radians =
  • (0.63418384082329 radians × 180 grades) / π radians =
  • 114.15309134819/π grades =
  • 36.336057514571 grades =
  • 36 grades, 20 minutes and 9.81 seconds

Calculate the measure of angle ∠B

  • m(∡B) =
  • arccos((42 + 32 - 62) / (2 × 4 × 3)) =
  • arccos((16 + (3 + 6) × (3 - 6))) / 24) =
  • arccos((16 + 9 × - 3)) / 24) =
  • arccos((16 - 27)) / 24) =
  • arccos( - 11/24) =
  • arccos(- 0.458333333333) =
  • 2.0469153877063 radians =
  • (2.0469153877063 radians × 180 grades) / π radians =
  • 368.44476978713/π grades =
  • 117.27961273595 grades =
  • 117 grades, 16 minutes and 46.61 seconds

Calculate the measure of angle ∠C

  • m(∡C) =
  • arccos((42 + 62 - 32) / (2 × 4 × 6)) =
  • arccos((16 + (6 + 3) × (6 - 3))) / 48) =
  • arccos((16 + 9 × 3)) / 48) =
  • arccos((16 + 27)) / 48) =
  • arccos(43/48) =
  • arccos(0.895833333333) =
  • 0.46049342505981 radians =
  • (0.46049342505981 radians × 180 grades) / π radians =
  • 82.888816510766/π grades =
  • 26.384329749451 grades =
  • 26 grades, 23 minutes și 3.59 seconds

Check if the sum of the angles ∡A + ∡B + ∡C = 180°

  • ∡A = 36.336057514571°
  • ∡B = 117.27961273595°
  • ∡C = 26.384329749451°

Note: The sum of the angles is approximately 180° instead of exactly 180°. This is because we performed calculations that required rounding: calculating arccos and converting from radians to degrees.


  • m(∡A) + m(∡B) + m(∡C) =
  • 36.336057514571° + 117.27961273595° + 26.384329749451° =
  • 179.99999999998° ≈ 180°


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