How this instrument works
Three side lengths are enough to fix a triangle's shape entirely, with no angle measured first. The Law of Cosines makes that fact usable: cos A = (b² + c² − a²) / (2bc) recovers the angle opposite side a directly from the three lengths, and swapping letters gives B the same way. This is the SSS case of triangle-solving — distinct from a solver that starts with two angles and the side between them, or one that starts with two angles and a side that is not between them.
The rearrangement comes from generalising the right-triangle case to any triangle at all. Picture the triangle sitting with side c along the bottom; the height dropped from the opposite vertex splits it into two right triangles sharing that height, and writing the ordinary sum-of-squares relation for each piece, then eliminating the shared height, leaves c² = a² + b² − 2ab cos C. Solve that single line for cos C, relabel the letters twice to reach the other two vertices, and the whole solver is that one algebra step performed three times.
Because cosine turns negative past 90°, the identical formula handles an obtuse triangle without any special case — a negative value under acos simply reports an angle wider than a right angle, something a bare ratio of two sides cannot do on its own. The other limit sits at the triangle inequality: as one side grows toward the sum of the other two, the angle opposite it stretches toward a flat 180°, and once that sum is exceeded no triangle exists to solve, which is precisely what this sheet's checks catch before returning a number.
- Enter the three known lengths into Side a, Side b, and Side c — any order, any consistent unit.
- Angle A (opposite a) and Angle B (opposite b) fill in first, straight from the Law of Cosines rearrangement above.
- Angle C (opposite c) follows by subtraction, since a triangle's three angles always total 180°.
- Switch an angle field's unit between degrees, radians, or turns to see the same solve reported differently.
- If the three lengths cannot close into a triangle, the sheet flags the input instead of returning a false angle.
Worked example — the 3-4-5 triangle solved from its sides alone
Feed the sheet Side a = 3, Side b = 4, and Side c = 5, with no angle supplied. Side a sits opposite Angle A, so cos A = (b² + c² − a²) / (2bc) = (16 + 25 − 9) / 40 = 0.8, giving A = acos(0.8) ≈ 36.8699° (0.64350111 rad). The same substitution for side b gives cos B = (9 + 25 − 16) / 30 = 0.6, so B = acos(0.6) ≈ 53.1301° (0.92729522 rad).
Angle C costs no further trigonometry: C = 180° − 36.8699° − 53.1301° = 90.0000° exactly, or π/2 ≈ 1.57079633 rad underneath. That right angle is no coincidence — 3² + 4² = 5² is the classic whole-number triple, and because the Law of Cosines is that very relation generalised, it lands on the identical 90° that a plain opposite-over-adjacent arctangent would give for the same triangle, confirming the two methods never disagree once a right angle is actually present.
Questions
What three measurements does this calculator need?
Only the three side lengths, a, b, and c — no angle at all. That is the SSS case of triangle-solving, distinct from a solver that starts with two angles and the included side, or one that starts with two angles and a side that is not between them.
Why does finding an angle require an inverse cosine?
Because the Law of Cosines first hands you cos A, not A itself — cos A = (b² + c² − a²) / (2bc) is a ratio. Acos is the operation that turns that ratio back into the angle producing it, the same role asin plays for a sine ratio.
How does this formula reduce to the Pythagorean relation?
Set c as the side opposite the angle being solved and test whether a² + b² equals c² exactly: in that one case the numerator of cos C is zero, so acos(0) hands back exactly 90°. A 3-4-5 triangle is the smallest whole-number example where this happens.
What is the most common mistake with this formula?
Pairing the wrong side with the wrong angle — a sits opposite A, not next to it, and swapping that pairing gives a wrong-but-plausible-looking angle. The reliable check afterward: A, B, and C should sum to exactly 180°.
Can any three lengths form a triangle?
No. Each side must be shorter than the sum of the other two — the triangle inequality. Lengths like 2, 3, and 10 fail immediately, since 2 + 3 falls far short of 10; this sheet tests all three combinations before reporting an angle.
Does the angle unit — degrees, radians, or turns — change the math?
No, only the display. The Law of Cosines is solved once, and the result is converted into whichever unit the field is set to afterward. A 3-4-5 triangle's right angle reads as 90° in degrees or exactly π/2 ≈ 1.5708 in radians — the same angle either way.