How this instrument works
An oblique triangle is simply one with no right angle anywhere in it — which means the ordinary Pythagorean theorem, built specifically for right triangles, doesn't apply. The Law of Cosines fills that gap: given two sides, a and b, and the angle C sitting between them, the third side works out to c = √(a² + b² − 2ab·cos(C)). It handles acute, obtuse, and right shapes alike, all with one formula.
That last case is worth sitting with. Set the included angle to exactly 90° and cos(90°) equals zero, so the subtracted term −2ab·cos(C) vanishes entirely and the formula collapses to c = √(a² + b²) — the Pythagorean theorem itself. A right-triangle-only tool elsewhere on this site solves that special case directly; this page is the general instrument standing behind it, valid whether the included angle is 90° or any other value at all.
The cosine term does real work as the angle changes. Past 90°, cosine turns negative, so −2ab·cos(C) turns positive and adds to a² + b² instead of trimming it — the third side stretches out longer than either known side alone would suggest for a wide, blunt included angle. Below 90°, cosine stays positive and keeps subtracting, pulling the third side in shorter than a plain sum of squares would give.
- Enter Side a and Side b — the two sides you already have measurements for.
- Enter Included angle C — specifically the angle trapped between sides a and b, not one opposite either side.
- Read Side c (opposite angle C) — the completed third side.
- Switch the angle's unit between degrees, radians, and turns; the field converts on the fly.
Worked example — sides 5 and 7 meeting at 60°
Side a = 5, Side b = 7, and Included angle C = 60°. Squares first: a² + b² = 25 + 49 = 74. Then 2ab = 2 × 5 × 7 = 70, and cos(60°) = 0.5, so 2ab·cos(C) = 70 × 0.5 = 35. Subtract: 74 − 35 = 39, and c = √39 ≈ 6.244997998398398.
Change the included angle to exactly 90° with sides 3 and 4, and the cosine term disappears entirely: c = √(9 + 16 − 0) = √25 = 5 — the ordinary 3-4-5 right triangle, produced by the same formula with no adjustment needed. Set both known sides to 10 with a 60° angle between them instead, and the third side comes back as exactly 10 too, since a 60° angle wedged between two equal sides always closes into an equilateral triangle.
Questions
What is the Law of Cosines formula?
c = √(a² + b² − 2ab·cos(C)), where a and b are two known sides and C is the angle enclosed between them. With a = 5, b = 7, and C = 60°, that works out to √(74 − 35) = √39 ≈ 6.245.
What makes a triangle oblique?
Simply having no right angle anywhere in it. Any triangle that isn't a right triangle qualifies — acute triangles, where every angle stays under 90°, and obtuse triangles, where one angle exceeds 90°, are both considered oblique.
Does the Law of Cosines still work on a right triangle?
Yes — set the included angle to 90° and cos(90°) equals zero, so the formula reduces exactly to c = √(a² + b²), the ordinary Pythagorean theorem. A right triangle is simply the one special case where this general formula happens to simplify.
Why must angle C sit between sides a and b specifically?
Because the formula is built around that exact arrangement — its derivation drops a perpendicular from the vertex opposite c straight through the enclosed angle. Feed in an angle that sits opposite one of the two known sides instead, and this formula no longer describes the triangle in front of you; that arrangement calls for the Law of Sines instead.
What happens to Side c when the included angle is obtuse?
Cosine turns negative past 90°, which flips the sign of the −2ab·cos(C) term into a positive addition. Side c ends up longer than √(a² + b²) alone would give — a visible sign that a wide, blunt angle is stretching the far side out.
Can this formula produce an equilateral triangle?
Yes — whenever the two known sides are equal and the included angle is exactly 60°. With a = b = 10 and C = 60°, the third side comes back as exactly 10 too, matching both of the others: every 60° angle wedged between two equal sides closes into an equilateral shape.