How to Find Angles & Sides of a Triangle

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Many math classes and standardized tests, such as the ACT and SAT, will require you to find a triangle’s angles and sides. Triangles can be categorized as right (having a 90-degree angle) or oblique (non-right); as equilateral (3 equal sides and 3 equal angles), isosceles (2 equal sides, 2 equal angles) or scalene (3 different sides, 3 different angles); and as similar (2 or more triangles that have all angles equal and all sides proportional). The strategy you use to find angles and sides depends on the type of triangle and the number of sides and angles you are given.

    Draw and label your triangle according to the information you are given.

    Try geometry before trigonometry. While you can use trig to find every side and angle, geometry is usually quicker and easier. First, remember the sum of the angles of any triangle is always 180 degrees. If you know 2 angles of a triangle, you can always subtract their sum from 180 to find the third angle. Every angle of an equilateral triangle is always 60 degrees. For isosceles triangles, it is important to remember that the two equal sides will face the two equal angles (so if angle A = angle B, side A = side B). For right triangles, remember the Pythagorean Theorem (the sum of the squares of the two shorter sides is equal to the square of the hypotenuse, or a² + b² = c² ). For similar triangles, remember that the sides of similar triangles are proportionate and solve using ratios (for example, the ratio of the first triangle’s side a and side b will be equal to the second triangle’s side a and side b).

    Use trigonometric ratios to find missing angles of right triangles. The three basic trig ratios are Sine = Opposite / Hypotenuse; Cosine = Adjacent / Hypotenuse; and Tangent = Opposite / Adjacent (often remembered with the mnemonic device “SohCahToa”). Solve for the missing angle by using the arcsin, arccos or arctan function of your calculator (usually labeled as “sin-1,” “cos-1” and “tan-1”). For example, to find angle A given that side a = 3 and side b = 4, since tanA = 3/4 , you would enter arctan(3/4) into your calculator to get angle A.

    Use the Law of Cosines and/or the Law of Sines to find missing angles and sides of oblique (non-right) triangles. You will need to use the Law of Cosines (c² = a² + b² - 2ab cosC) if you are given 3 sides and 0 angles, or if you are given two sides and the angle opposite the missing side. The Law of Sines (a/sinA = b/sinB = c/sinC) can be used any time you know the length of one side and its opposite angle and one other side or angle.

    Check your answers. Remember the shortest side will face the shortest angle, and the longest side will face the longest angle (so if side a < side b < side c, then angle A < angle B < angle C). Another way to check your results is the Triangle Inequality Theorem, which states that any side of a triangle must be greater than the difference of the other two sides and less than the sum of the other two sides.

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