How this instrument works
The Law of Cosines generalizes the Pythagorean theorem to work for any triangle, not just right triangles. Given all three side lengths, it recovers the angle opposite any chosen side: C = acos((a² + b² − c²) ⁄ 2ab), where C is the angle opposite side c. This is the reverse direction of a full angle-and-side (SAS) solver — instead of computing a missing side from two sides and their included angle, this finds a missing ANGLE from all three already-known sides.
Set the angle to exactly 90° and rearrange, and a² + b² − c² becomes exactly 0, leaving a² + b² = c² — the ordinary Pythagorean theorem, revealed as the special case of the Law of Cosines where the triangle happens to have a right angle. The full law extends that same relationship to acute and obtuse triangles as well, adding a correction term that vanishes precisely at 90°.
Because SSS (side-side-side) is a full triangle-congruence condition, three known sides always determine a triangle's shape uniquely — meaning every one of its three angles can be recovered this same way, applying the formula three times with a different side playing the role of c each time.
- Enter the triangle's three side lengths into the Side a, Side b, and Side c fields.
- Read Angle C: the sheet computes the angle opposite side c using the Law of Cosines.
- To find a different angle, relabel which side is called c and re-enter the values in that order.
Worked example — sides 5, 7, and 8
A triangle has sides 5, 7, and 8. The angle opposite the 8-unit side is C = acos((25+49−64) ⁄ 70) = acos(10⁄70) ≈ 81.79° — recovered entirely from the three side lengths, with no separate angle measurement needed at all.
A 3-4-5 right triangle, with c set to 5 (the longest side, the hypotenuse), gives C = acos((9+16−25) ⁄ 24) = acos(0) = 90° exactly — the Law of Cosines correctly reducing to a right angle whenever the classic Pythagorean relationship a² + b² = c² already holds.
Questions
What is the Law of Cosines?
C = acos((a² + b² − c²) ⁄ 2ab), a formula that recovers the angle opposite side c from all three of a triangle's known side lengths. It generalizes the Pythagorean theorem to work for any triangle, not just right triangles.
How does the Law of Cosines relate to the Pythagorean theorem?
Setting the angle to exactly 90° makes a² + b² − c² equal exactly 0 inside the formula, which is precisely the Pythagorean theorem rearranged. The Law of Cosines is a full generalization that still agrees with the Pythagorean theorem at that special right-angle case.
Can I find all three angles of a triangle from its three sides?
Yes — since three sides determine a triangle's shape completely (the SSS congruence condition), applying this same formula three times, relabeling which side plays the role of c each time, recovers all three angles.
How is this different from a combined SAS solver?
An SAS solver goes the opposite direction: given two sides and the angle between them, it finds the third side and the remaining angles. The Law of Cosines used here instead starts from all three sides already known and recovers an angle — the reverse problem.
What if the three sides can't form a real triangle?
The Triangle Inequality must hold (any two sides summing to more than the third) for the formula's argument to stay within the valid range for arccosine, −1 to 1. If it fails, no real triangle exists with those sides, and this calculator flags the input.