How this instrument works
The Law of Cosines answers exactly one question: given two sides of a triangle and the angle trapped between them, how long is the side that closes the shape? That side-angle-side combination pins a triangle down completely — there is only one possible third side for a given a, b, and C — so c² = a² + b² − 2ab·cosC is not an approximation but an exact identity, extending the Pythagorean theorem to triangles that need not have a right angle anywhere in them.
The extra term, −2ab·cosC, is doing the work of a correction. Picture angle C sitting at the origin with sides a and b radiating out from it as two vectors; the missing side c is the straight-line gap between their far ends, and the Law of Cosines is just the vector identity |a − b|² = a² + b² − 2a·b written out in full, since a dot product a·b equals ab·cosC by definition. Set C to a right angle and that dot product vanishes — cos90° = 0 — so the whole correction term disappears, leaving the plain a² + b² = c² behind.
Euclid proved the obtuse and acute cases geometrically in Book II of the Elements, propositions 12 and 13, centuries before trigonometric notation existed to write cosC at all; the Persian astronomer Ghiyath al-Kashi later generalized and tabulated the relationship in the fifteenth century, which is why French textbooks still call it le théorème d'Al-Kashi. At the formula's extremes the geometry stays honest: as C opens toward a straight 180°, cosC slides toward −1 and c climbs toward a + b, the triangle flattening into a line; as C closes toward 0°, cosC approaches 1 and c shrinks toward the gap a − b — the formula enforcing the triangle inequality on its own, with nobody stating that rule separately.
- Enter the two known side lengths into Side a and Side b, in any consistent unit — the third side comes back in that same unit.
- Enter the angle trapped between those two sides into Included angle C, picking degrees, radians, or turns from its unit menu.
- Confirm C is the angle where sides a and b actually meet, not one of the triangle's other two corners, since the formula only works with the enclosed angle.
- Read the missing length straight off Side c (opposite C).
Worked example — 5 and 7 enclosing 60°
Take a triangle with Side a = 5 units, Side b = 7 units, and an Included angle C of 60°, or 1.0471975511965976 radians once converted. The Law of Cosines gives c² = 5² + 7² − 2×5×7×cos(60°) = 25 + 49 − 70×0.5 = 74 − 35 = 39, so Side c (opposite C) = √39 = 6.244997998398397, the exact figure this sheet returns.
Picture that triangle as a plot of land: two fence runs of 5 m and 7 m meet at a corner post bent to 60°, and the survey wants the length of the third run that closes the boundary. Nobody has to walk out and measure it — √39 m ≈ 6.245 m is fixed the moment the two runs and the angle between them are fixed, which is the entire promise of a side-angle-side triangle: three numbers in, one exact triangle out.
Questions
What does the Law of Cosines actually compute?
The length of a triangle's third side from the other two sides and the angle trapped between them: c² = a² + b² − 2ab·cosC. It is the direct extension of the Pythagorean theorem to triangles without a right angle — set C to exactly 90° and the −2ab·cosC term vanishes, leaving the familiar a² + b² = c² behind.
How is this different from the ASA and SSS triangle solvers?
Each starts from a different known trio. This one takes side-angle-side — two sides and the angle squeezed between them — and solves directly for the third side with the Law of Cosines. ASA starts from two angles and the side between them instead, solving with the Law of Sines; SSS starts from all three sides and works backward for the angles. Whichever measurements you already have decides which of the three applies.
Why does a 90° included angle give the same answer as the Pythagorean theorem?
Because cos(90°) equals zero, so the correction term −2ab·cosC disappears entirely and c² = a² + b² − 2ab·cosC reduces to plain c² = a² + b². Sides of 3 and 4 enclosing a right angle return c = 5 exactly — the Pythagorean theorem was never a separate rule, just the Law of Cosines at its one special angle.
Does a wider included angle always make the third side longer?
Yes, for fixed side lengths — cosine falls as an angle opens from 0° toward 180°, so −2ab·cosC grows and c grows with it. Two 5-unit sides enclosing 120° close to a third side of 5√3 ≈ 8.660, longer than the same two sides would give at a right angle; push C toward 180° and c climbs all the way toward the flat-line limit of a + b.
Which angle counts as the included angle C?
Specifically the one where sides a and b physically meet, not either of the triangle's other two corners. Plugging in a non-included angle still returns a number from the formula, but it will not be the correct third side, because the −2ab·cosC term is derived from the vertex where a and b actually join.
Can the Law of Cosines produce an impossible triangle?
No — given any positive a, b and any C strictly between 0° and 180°, the formula always returns a valid c that satisfies the triangle inequality on its own. As C approaches 180° the answer approaches a + b, and as C approaches 0° it approaches the gap a − b — the two boundaries a real triangle can never reach — so every output between them is guaranteed constructible.