A triangle is a three-sided polygon. Instructors often ask intermediate and advanced-level math students to calculate the missing angle in a triangle. One method of finding a missing angle is based on the premise that the sum of the interior angles of a triangle equals 180 degrees. Another approach involves using a formula based on the trigonometric sine rule. When solving such problems, the number of known angles in the triangle determines the method you must use.

## When Two Angles are Given

There is no need to perform any calculations if asked to find one or more angles in an equilateral triangle, or one whose three sides are of equal length. The angles of an equilateral triangle are always equal to 60 degrees.

Add the two known angles together when working with a triangle for which two of the angles are given.

Find the missing angle by subtracting the sum of the two angles from 180.

Express the answer in degrees.

Use the sine rule if given only one angle and two lengths of a triangle. The formula is sin A/a = sin B/b, where “A” and “B” are angles and “a” and “b” are the lengths of the sides opposite these angles, respectively.

Suppose that you are solving a triangle for which one angle equals 25 degrees and the side opposite this angle measures 7 units. An adjacent angle, A, is opposite a side measuring 12 units. Plugging these numbers into the formula would provide: sin(A)/12 = sin(25)/7. Rearranging this equation results in sin(A) = sin(25)*12/7. Using a scientific calculator to find sin(25), carrying out the rest of the equation would show that sin(A) = 0.724. To find angle “A,” use the calculator to determine the inverse sine of 0.724. The answer is approximately 46 degrees.

Keep in mind that inverse sine yields two solutions; your calculator will only give you one of these solutions. Examine the angle you were asked to find. If it is obtuse, it measures more than 90 degrees. If you are unsure whether the angle is obtuse or acute, measure it with a protractor. In the example used here, angle A is obtuse; it cannot equal 46 degrees, as suggested by the original solution. Subtract 46 from 180 to get the correct solution, 134 degrees.

Use the method described in the previous section to find the remaining angle.

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