Triangle Calculator

Solving a Triangle: What It Means

"Solving" a triangle means finding every side length and every angle measure once you already know a valid combination of three of those six values. This is different from computing the area of a triangle whose sides are already fully known (covered on the Area & Volume Calculator's triangle option using Heron's formula) — here, some sides and angles are unknown, and the goal is to recover them using the Law of Sines and Law of Cosines.

Law of Cosines (for SSS and SAS)

When all three sides are known (SSS), or two sides and the angle between them (SAS), the Law of Cosines finds the remaining piece directly:

$$c^2 = a^2 + b^2 - 2ab\cos(C)$$

For SSS, the formula is rearranged to solve for each angle instead of each side. For SAS, it's used directly to find the third side, and then the Law of Sines finds the remaining two angles.

Law of Sines (for ASA and AAS)

When two angles and a side are known (ASA or AAS), the Law of Sines relates every side to the sine of its opposite angle:

$$\frac{a}{\sin(A)} = \frac{b}{\sin(B)} = \frac{c}{\sin(C)}$$

The third angle always follows immediately, since all three angles in any triangle sum to 180 degrees, and the remaining sides follow from the ratio above once every angle is known.

Worked Example: SAS

Given side a equal to 5, side b equal to 7, and the included angle C equal to 45 degrees: the Law of Cosines gives side c squared equal to 25 plus 49 minus 2 times 5 times 7 times the cosine of 45 degrees, which works out to c approximately 4.95. From there, the Law of Cosines (applied again, this time solving for an angle) finds Angle A at roughly 45.6 degrees and Angle B at roughly 89.4 degrees (the three angles summing to 180, as they always must). The area, using one-half times a times b times the sine of the included angle, comes out to about 12.37 square units.

Validating a Real Triangle

Not every combination of three numbers describes an actual triangle. For SSS, each side must be strictly shorter than the sum of the other two (the triangle inequality) — three sides of length 2, 2, and 5 can never close into a triangle, since 2 plus 2 doesn't even reach 5. For any case involving angles, the two given angles must sum to less than 180 degrees, leaving a positive amount for the third. This calculator checks both conditions before attempting a solution and reports a clear error instead of a nonsensical result if either fails.

Common Triangle-Solving Mistakes

Mixing up which angle is "included" between two given sides is a frequent SAS error — the Law of Cosines specifically needs the angle between the two known sides, not some other angle in the triangle. Forgetting to convert between degrees and radians when working by hand (this calculator handles that conversion internally) is another common source of wrong answers. For ASA and AAS, confusing which side is opposite which angle when applying the Law of Sines can silently swap two results that look individually plausible but don't actually belong to the same triangle.

Triangle-Solving Terms You Should Know

Law of Cosines — relates one side of a triangle to the other two sides and the angle between them; a generalization of the Pythagorean theorem to non-right triangles.

Law of Sines — states that the ratio of each side to the sine of its opposite angle is the same for all three sides of a triangle.

Included Angle — the angle formed directly between two given sides, required for the SAS case.

Triangle Inequality — the rule that each side of a triangle must be shorter than the sum of the other two, or no triangle can be formed.