How to Solve Any Triangle, Case by Case

What it means to solve a triangle

Every triangle has six parts: three sides and three angles. In general, knowing three of them, including at least one side, fixes the triangle, and the other three can be calculated. Knowing only the three angles is not enough, because similar triangles of every size share the same angles; that case is sometimes called AAA and has no unique solution.

The known parts are described by the order in which they appear around the triangle. ASA means two angles with the side between them; SAS means two sides with the angle between them. The six possible patterns, and the tool for each, are:

• Right triangle with any two other parts: basic ratios and the Pythagorean theorem.
• AAS or ASA: angle sum, then the law of sines.
• SAS: the law of cosines for the third side, then the remaining angles.
• SSS: the law of cosines for the angles.
• SSA: the law of sines with a check for zero, one or two triangles.
• AAA: shape only, no unique size.

Always sketch the triangle and label the given parts before choosing a method. Most errors come from pairing a side with the wrong angle.

Right triangles: SOH CAH TOA

In a right triangle, for an acute angle $\theta$, the three basic ratios relate the angle to the sides: the opposite side, the adjacent side and the hypotenuse.

To find an angle, use an inverse ratio. A 3, 4, 5 right triangle has its smaller acute angle at $\tan^{-1}\frac{3}{4} \approx 36.87^\circ$.

Two angles and a side (AAS, ASA)

Find the third angle by subtracting from $180^\circ$. You then have every angle and one side, which is a complete pair for the law of sines:

These cases always have exactly one solution, provided the two given angles add to less than $180^\circ$.

Two sides and the included angle (SAS)

Use the law of cosines to find the side opposite the known angle. Then find a second angle. The safest choice is the angle opposite the shorter of the two original sides, because it must be acute, so the inverse sine gives it without ambiguity.

Three sides (SSS)

First confirm the sides satisfy the triangle inequality: each side must be shorter than the sum of the other two. Then use the cosine form of the law of cosines, starting with the angle opposite the longest side:

Two sides and a non-included angle (SSA)

This is the ambiguous case. Solve for the unknown angle with the law of sines, $\sin B = \frac{b\sin A}{a}$, and then examine the result:

• If $\sin B > 1$, no triangle exists.
• If $\sin B = 1$, there is one right triangle.
• If $\sin B < 1$, test both $B$ and $180^\circ - B$. Each one that leaves a positive third angle gives a valid triangle.

For example, $A = 30^\circ$, $a = 6$ and $b = 10$ give $\sin B = \frac{5}{6}$, so $B \approx 56.44^\circ$ or $123.56^\circ$. Both leave room for a third angle, so there are two triangles. The law of sines article works both of them out in full, along with the no-triangle and one-triangle cases.

Angles of elevation and depression

Many applied problems describe angles measured from the horizontal. An angle of elevation is measured upwards to an object; an angle of depression is measured downwards. Because the horizontal lines at the observer and at the object are parallel, the angle of depression from A to B equals the angle of elevation from B to A.

Problems with two observations, such as sighting the top of a building from two points along the ground, are solved by combining a right triangle with an oblique one, often using the law of sines in the oblique part.

A checklist for every answer

• The three angles add to $180^\circ$.
• The longest side is opposite the largest angle, and the shortest side opposite the smallest.
• Every side is shorter than the sum of the other two.
• Recomputing one side with the other law gives the same value, up to rounding.
• Answers are rounded only at the end, to the precision the question asks for.

Once a triangle is solved, its area follows from $\frac{1}{2}ab\sin C$ using any two sides and the angle between them. The law of cosines calculator solves all the oblique cases with steps, and the distance formula calculator handles triangles given by coordinates.

Rounding and precision

Triangle problems chain several calculations together, so small rounding errors early on can grow into visible differences at the end. Keep full calculator precision for every intermediate value, either by storing results in memory or by writing at least four significant figures. Round only the final answers, and round them to the precision the question asks for, usually one or two decimal places for angles in degrees and a sensible number of significant figures for lengths.

Because of rounding, the three angles you report may add to $179.9^\circ$ or $180.1^\circ$. That is not an error in method. If the sum is off by a whole degree or more, though, something has gone wrong, most often a side paired with the wrong angle or a missed obtuse solution. A second independent calculation, such as finding the last side with both laws, is the most reliable way to confirm that your answer is right.

Frequently asked questions