Triangles are one of the most prevalent shapes in plane geometry. Not only do they occur frequently as independent shapes, but they also form parts of more complex shapes. Therefore, it is important to know how to find the lengths of the sides of a triangle. There are a few ways in which this can be done, depending on the type of triangle and the information you have.

Determine whether your triangle is a right triangle. A right triangle consists of one 90 degree angle and two others. If you have a right triangle, the Pythagorean Theorem applies. The Pythagorean Theorem states that a right triangle's sides can be defined by the following equation:

a^2 + b^2 = c^2,

"C" is the triangle's hypotenuse (the side opposite the 90 degree angle) and "a" and "b" are the other two sides.

Therefore, if you know two sides of the triangle, you can use them to find the third.

Enter the two known sides into the Pythagorean Theorem. If one of the sides is the hypotenuse, make sure to substitute that value in for "c." For example, if you know that the hypotenuse is "26" and another side is "10," plug those values in for "c" and "a," respectively:

10^2 + b^2 = 26^2

Solve for the missing side. For example, if a = 10 and c = 26:

10^2 + b^2 = 26^2

100 + b^2 = 676

b^2 = 576

b = 24

Check your right triangle's angle measurements. If the triangle's angles are 30, 60 and 90 or 45, 45 and 90, it is a special triangle; its sides stand in a specific proportion with one another. In a 30/60/90 triangle, if the side opposite the 30 degree angle is X, the side opposite the 60 degree angle is X times the square root of 3, and the hypotenuse is X times 2. For example, if the side opposite the 30 degree angle is 4, the side opposite the 60 degree angle is 4 times the square root of 3, and the hypotenuse is 8. In a 45/45/90 triangle, if the side opposite either 45 degree angle is X, the hypotenuse is X times the square root of 2. For example, if the side opposite a 45 degree angle is 3, the side opposite the other 45 degree angle is also 3, and the hypotenuse is 3 times the square root of 2. Memorising these special proportions can save you the work of using the Pythagorean Theorem.

Record the known angles and sides of your triangle. If you know either two angles and one side, or two sides and one of the angles that is not surrounded by those sides, you can use the law of sines to calculate the missing side. The law of sines states that, for any triangle: a/sin(A) = b/sin(B) = c/sin(C), where a, b and c are the sides of the triangle, and A, B and C are the angles opposite those sides.

Enter the information you have into the law of sines. For example, for a= 5 , b = 7, and angle B = 50 degrees, write the following:

5/sin(A) = 7/sin(50)

Isolate and solve for the missing piece. In the example,

5/sin(A) = 7/sin(50)

so,

sin(A)/5 = sin(50)/7

sin(A) = 5sin(50)/7

sin(A) ~= 0.54717

A ~= 33 degrees

Determine the measurement of the third angle. Since the sum of all interior angles of a triangle is 180 degrees, A + B + C = 180. For example,

33 + 50 + C = 180

88 + C = 180

C = 92

Plug the value for the third angle into the law of sines, and use one of the already-existing ratios to solve for the third side. For example:

7/sin(50) = c/sin(92)

c = 7sin(92)/sin(50) ~= 9.133

If you know two angles and one side, solve for the third angle first, and then solve for the other two sides using the known side-to-angle ratio.