How to Find an Angle in a Triangle

Why do so many students get stuck on a missing triangle angle? Usually they reach for a formula before checking what the problem actually gives them. There are only five starting situations, and each one points to a single method. Once you match your givens to the right tool, every angle takes two or three short steps.

How Do You Pick the Right Method?

Pick the method by listing what you already know: angles, sides, and whether one angle is 90 degrees. That list decides everything. Mathematicians label the cases with letters, where A stands for a known angle and S for a known side.

The five common cases are AA (two angles), a right triangle with two sides, SSS (three sides), SAS (two sides with the angle between them), and SSA (two sides with an angle that is not between them). The tree below maps each case to its method.

One case never works: three angles with no sides. Angles alone fix the shape but not the size, so the side lengths stay unknown.

How Do You Find an Angle When You Know Two Angles?

Subtract the two known angles from 180 degrees. Every triangle’s interior angles add to exactly 180, so the third angle is whatever remains. With angles of 47 and 68 degrees, the third is 180 minus 115, or 65 degrees.

Isosceles and Equilateral Shortcuts

An isosceles triangle has two equal sides, and the angles opposite them are also equal. Given a 40 degree apex angle, each base angle is (180 minus 40) divided by 2, which is 70 degrees. Given a 70 degree base angle instead, the apex is 180 minus 140, or 40 degrees.

An equilateral triangle needs no work at all. Its three angles are always 60 degrees each, whatever the side length.

Exterior Angles

An exterior angle forms when you extend one side past a corner. It equals the sum of the two interior angles at the other corners. With interior angles of 50 and 60 degrees, the exterior angle at the third corner is 110 degrees.

That exterior angle and its neighboring interior angle form a straight line, so they add to 180. Here the interior angle is 70 degrees, which matches the sum rule.

How Do You Find an Angle in a Right Triangle?

Use an inverse trig function on the ratio of two known sides. Choose the ratio by which sides you have relative to the angle you want: opposite, adjacent, or hypotenuse.

The other acute angle is 90 minus the first. For the 5-12-13 triangle, that is 90 minus 22.62, or 67.38 degrees. The ratios themselves come from sine, cosine, and tangent, which our guide on sine, cosine, and tangent explained covers from the ground up.

Check your calculator mode before pressing the key. In radian mode, asin(5/13) returns 0.3948 instead of 22.62, and the degrees vs radians guide shows how to convert between the two.

How Do You Find Angles When You Know All Three Sides?

Use the law of cosines, solved for the angle: cos C = (a^2 + b^2 – c^2) / 2ab. Here C is the angle opposite side c. Repeat with each side in the c position to get all three angles.

Take a triangle with sides 5, 6, and 8. For the angle opposite 8, the top of the fraction is 25 + 36 – 64, which is -3. The bottom is 2 x 5 x 6, which is 60. So cos C = -0.05, and C = 92.87 degrees.

The same steps give 38.62 degrees opposite side 5 and 48.51 degrees opposite side 6. The three angles total 180.00, which confirms the work.

A negative cosine is a useful signal. It means the angle is obtuse, above 90 degrees, and inverse cosine handles it correctly. Inverse sine cannot, which is why SSS always uses cosines. To run any side set instantly, the triangle angle calculator for three sides or two angles returns all three angles and the triangle type.

Check Your Answer Two Ways

First, the three angles must add to 180 degrees, allowing about 0.01 for rounding. Second, the largest angle must sit opposite the longest side, and the smallest opposite the shortest. In the 5-6-8 triangle, 92.87 sits opposite 8 and 38.62 opposite 5, so both checks pass.

What If You Know Two Sides and the Angle Between Them?

Find the third side with the law of cosines, then find a second angle with the law of sines. The third angle comes from the 180 rule.

Take sides b = 10 and c = 12 with the included angle A = 40 degrees. The law of cosines gives a^2 = 100 + 144 – 240 cos 40, which is 60.15. So side a is 7.76.

Next apply the law of sines, a / sin A = b / sin B. Then sin B = 10 x sin 40 / 7.76, which is 0.8288, so B = 55.98 degrees. Finally C = 180 – 40 – 55.98, which is 84.02 degrees.

Here is the practical insight. Always use the law of sines on the angle opposite the shorter unknown side. That angle cannot be obtuse, so inverse sine gives the true value. Solving for the angle opposite 12 first would risk picking the wrong one of two sine answers. A law of cosines check on C returns 84.02 as well.

What Is the Ambiguous Case With Two Sides and a Non-Included Angle?

The SSA case uses the law of sines, but the givens can fit zero, one, or two different triangles. You must test which situation you have before trusting the answer.

Take angle A = 35 degrees, side a = 8 opposite it, and side b = 12. The law of sines gives sin B = 12 x sin 35 / 8, which is 0.8604. Inverse sine returns B = 59.36 degrees.

Sine is positive for both an angle and its supplement, so B = 180 – 59.36 = 120.64 also works. Each value is valid when it leaves room for a positive third angle. Here, C = 85.64 in the first triangle and 24.36 in the second.

How to Count the Triangles

Compare side a with the height h = b sin A, which is 12 x sin 35, or 6.88. Shorter than 6.88 means no triangle; for a = 6, sin B works out to 1.15, which is impossible. Equal to h gives one right triangle. Between h and b gives two triangles, as with a = 8. At least as long as b gives exactly one.

FAQs About Finding Triangle Angles

Sources

This article teaches plane triangle methods with angles measured in degrees. Spherical triangles on a globe follow different rules. Reviewed for accuracy by Prof. Dr. Khalil Mudassar, PhD. Last updated September 26, 2026.