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Mollweide's Formula Checker

Enter the three sides and three angles to test Mollweide's formula.

How it is calculated

Mollweide's identity says that in any triangle (a − b) ÷ c = sin((A − B) ÷ 2) ÷ cos(C ÷ 2), where a, b, c are side lengths and A, B, C are the opposite angles, entered in degrees. Side a must face angle A, side b angle B and side c angle C. The tool computes the left side and the right side to four decimals and calls the triangle Consistent if they differ by less than 0.01, otherwise Inconsistent. Side c must be greater than zero. The tool does not check separately that the angles add to 180°; a wrong angle simply makes the two sides disagree, and an angle C near 180° makes the right side explode.

Worked example

Take the 3-4-5 right triangle with a = 4, b = 3, c = 5, and angles A = 53.13°, B = 36.87°, C = 90°. Left side: (4 − 3) ÷ 5 = 0.2. Right side: half of A − B is 8.13°, whose sine is about 0.1414; half of C is 45°, whose cosine is about 0.7071; 0.1414 ÷ 0.7071 is about 0.2. The tool shows 0.2000 on both sides and the verdict Consistent.

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Common mistakes

  • Pairing a side with the wrong angle. Side a must be opposite angle A; swapping labels for only one pair makes a correct triangle look inconsistent.
  • Entering angles in radians. The fields are in degrees, so 0.9273 rad must be typed as 53.13.
  • Rounding the angles heavily and then trusting a mismatch. The tolerance is 0.01, so whole-degree angles of 53 and 37 still pass here, but a bigger rounding error can fail.
  • Treating Consistent as a proof. It is a check on one identity with a tolerance; two sides that are nearly right can still pass, so also confirm that the angles add up to 180°.
  • Using an angle C close to 180°. The cosine of half of C is then near zero and the right side becomes a huge number, which signals a degenerate triangle rather than a calculator error.

About this calculator

Mollweide's formula is an identity that relates all six parts of a triangle — its three sides and three angles — in a single equation: (a−b)/c = sin((A−B)/2) / cos(C/2). Because it involves every side and angle at once, unlike the Law of Sines or Law of Cosines which use only a subset, it's especially useful as an independent check: if a triangle's sides and angles are correct, this identity must hold exactly.

Trigonometry students and teachers use Mollweide's formula to verify that a solved triangle (from the Law of Sines or Law of Cosines) is internally consistent, catching arithmetic or angle-ambiguity errors that a single-relationship check might miss. Surveyors and navigators solving oblique triangles have historically used it the same way, as a cross-check before trusting a triangulation result.

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Comments (1)

  • Editorial team

    Know what this check cannot see. The left side is a ratio, so it does not change if every side is scaled by the same factor: the 4-3-5 triangle and a 40-30-50 version both give 0.2000 and Consistent. A unit slip applied to all three sides will therefore pass, and the angles can still hide an error as long as the difference A minus B stays right. Compare at least one side against your measurement. What kind of mismatch have you been trying to catch?

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