How to Figure Out What an Acute Angle Equals in Degrees

Different kinds of triangle have different properties.

Geometry and trigonometry have a host of ways to determine the angle measurements of a triangle. You may need to use the Pythagorean theorem, general properties of triangles, or properties of specific kinds of triangles in order to find the answer. Solving these kinds of problems is a matter of writing down everything you know about the problem and then deducing everything you can about that problem, proceeding from the most obvious deductions to the most recondite.

Instructions

1. Convert any radian measures to degrees. 1 radian = (180 / pi) degrees. For example, if the acute angle is equal to (pi / 3) radians, then multiply the conversion equation by pi, yielding 1 pi radian = 180 degrees. Divide by 3 to get (1 pi radian) / 3 = 60 degrees, then factor the radian out of the numerator, yielding (pi / 3) radians = 60 degrees. Many high school trigonometry equations are just radian-degree conversions in disguise, so make sure to do this first.

2. Use the Pythagorean theorem if you know two sides of the triangle. If you know the two shorter sides, A and B, then A^2 + B^2 = C^2. If you know one of the shorter sides and the longest side, then C^2 - A^2 = B^2 or C^2 - B^2 = A^2.

3. Use the angles in isolation to find the length of the acute angle. If you are given two angles besides the acute angle, then add those angles together and subtract the sum from 180 to find the third angle. If you are given one angle (call it N), and you know that the other two angles are equal, then take (180 - N) / 2 to find the measures of the other two angles.

4. Use the sides of the triangle to find the measure of the acute angle. For a right triangle, if the shortest side of the triangle is half the length of the hypotenuse, then the angles of the triangle are 30, 60, and 90 degrees. If two sides of the triangle are equal, then their opposite angles are equal as well, so use the same process as used in Step 1 to find the other two angles.

Tips & Warnings

• The sum of the interior angles of a triangle is always 180 degrees.

• The sine of an angle is the opposite over the hypotenuse; the cosine is the adjacent over the hypotenuse; the tangent is the opposite over the adjacent.

• If two sides of a triangle are equal, their opposite angles are equal.

• If side A is greater than side B, then the opposite angle of A is greater than the opposite angle of B.

References

• West Texas University: Beginning Algebra Tutorial 33: Basic Geometry
• Clark University: Measurement of Angles
• Clark University: Right Triangles
• University of Washington: Properties of an Isosceles Triangle