How Do You Find the Side of a Triangle?

To find the side of a triangle, you must first know what type of triangle you are working with. A right triangle has a 90-degree angle as one of its three corners. With this triangle, you can invoke the Pythagorean theorem to determine a side of the triangle. A helpful rule for finding the sides of a nonright triangle is that the sum of the angles inside a triangle is 180 degrees. This rule is the first step in gaining enough information to use the rule of sines and rule of cosines.

Things You'll Need

• Calculator

Instructions

Right Triangle: Two Sides Known

1. Start off solving for the side of a right triangle, given two sides’ lengths, by squaring these two lengths.

2. Plug these two values into the Pythagorean theorem: A^2 + B^2 = C^2. Here, C is the longest side, or the hypotenuse, and the carets ^ indicate exponentiation. Enter the square of the hypotenuse for C^2, if you know it.

3. Solve for the hypotenuse, C, if that’s the unknown, by taking the square root of A^2 + B^2. If A is your unknown side, solve for the unknown side adjacent to the right angle by taking the square root of C^2 - B^2.

Right Triangle: One Side and One Nonright Angle Known

1. Solve for side A opposite the known nonright angle ? by using the equation H sin ? = A, where H is the length of the hypotenuse. Alternately, if A is known but H isn’t, use H = A / sin ?.

2. Solve for side A adjacent to (touching) the known nonright angle ? by using the equation H cos ? = A. Alternately, if A is known but H isn’t, use H = A / cos ?.

3. Use A tan ? = B to solve for side B opposite angle ? if both A and B are nonhypotenuse sides.

Nonright Triangle: Three of Six Measurements Known

1. Use the rule that the three angles of a triangle sum to 180 degrees if you know two angles already. You also need to know at least one side length.

2. Use the law of sines to solve for more angles or sides. The rule of sines states that A / sin a is constant all around the triangle, where A is the length of a side and a is the angle opposite that length. <br /><br />For example, if the two angles adjacent to a side of length 2 are 30 and 45 degrees, then, by step 1, the remaining angle is 105. So 2 / sin 105 is the constant value around the triangle of the ratio of a side to the sine of its opposite angle: 2 / sin 105 = 2.07. Side A opposite the 30-degree angle is found from A / sin 30 = 2.07, or A = 1.035.

3. Use the rule of cosines if you know the length of two sides and the angle between them. The rule of cosines says A^2 = B^2 + C^2 - 2BC cos a, where a is opposite A.<br /><br />For example, if the side-angle-side measurements are 2, 30 degrees and 3, then the remaining side must be the square root of 2^2 + 3^2 - (2 x 3 x cos 30) = 7.80. The square root is 2.79.

Tips & Warnings

• The rule or sines and rule of cosines apply regardless of which side you label A, B or C. For angles a, b and c and sides A, B and C, knowing three out of six for a nonright triangle will be enough to solve the equation--as long as at least one is a side length.<br /><br />The rule of cosines helps when you know an angle and the two adjacent sides (side-angle-side, sometimes written as SAS). The rule of sines can solve a triangle when a side and its opposite angle are known, along with one of the remaining four pieces of information.

References

• CSG Network: Right Triangle Angle and Side Calculator
• Teacher’s Choice: How to Solve a Triangle Given Two Angles and One Side