Triangle Side Calculator

Label each angle with a capital letter and the side opposite it with the matching lowercase letter. Then the given information decides the method:

Existence conditions. Angles must sum to exactly $180°$, and every side must satisfy the triangle inequality: $a + b > c$ for all three pairings. If a computed side or angle violates either, no such triangle exists — a valid answer, not a mistake.

Law of cosines — works for every triangle and reduces to Pythagoras when $C = 90°$:

$c^2 = a^2 + b^2 - 2ab\cos C \qquad \Longleftrightarrow \qquad \cos C = \frac{a^2 + b^2 - c^2}{2ab}$

The rearranged form is safe for finding angles because $\arccos$ returns values across the full $0°$ to $180°$ range, so an obtuse angle comes out correctly signed.

Law of sines — needs a complete side–angle pair:

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

The ambiguous case. Given SSA, $\arcsin$ only ever returns an acute angle, but $\sin\theta = \sin(180° - \theta)$. So after computing $B$, also test $B' = 180° - B$: if $A + B' < 180°$, a second valid triangle exists and both must be reported. When $\sin B > 1$, no triangle exists.

Safe habit: solve for the largest unknown angle with the law of cosines, then use the law of sines for the rest — that avoids the ambiguity entirely.

• Reporting one answer in the SSA case. Always test the obtuse partner $180° - B$ before deciding.
• Calculator in the wrong mode. Degrees and radians produce completely different sides; $\cos 52 \approx 0.616$ in degrees but $\approx 0.469$ in radians.
• Misusing the law of cosines as $c^2 = a^2 + b^2 - 2ab\cos C$ with the wrong angle. $C$ must be the angle between $a$ and $b$.
• Applying Pythagoras to a non-right triangle. It is only the $C = 90°$ special case of the law of cosines.
• Rounding intermediates. Carry full precision and round once at the end; rounding $\cos 52°$ to $0.62$ shifts $c$ in the second decimal.
• Pairing a side with the wrong angle. In the law of sines, $a$ must be opposite $A$.