How to Find a Missing Side of a Triangle
Finding a missing side of a triangle depends on the information you have. For right triangles, the Pythagorean theorem is the simplest approach. For oblique (non-right) triangles, you can use the Law of Cosines or the Law of Sines depending on the known values.
Methods for Finding a Missing Side
Pythagorean Theorem
Works only for right triangles. Relates the two legs to the hypotenuse.
Law of Cosines
Works for any triangle when you know two sides and the included angle.
Law of Sines
Works when you know one side and two angles of the triangle.
When to Use Each Method
Pythagorean Theorem
Use this when you have a right triangle and know two of the three sides. The hypotenuse is always the longest side, opposite the 90-degree angle. If you know the two legs (a and b), compute c = sqrt(a² + b²). If you know one leg and the hypotenuse, compute the other leg as a = sqrt(c² - b²).
Law of Cosines
Use this when you know two sides and the angle between them (SAS condition). It generalizes the Pythagorean theorem to all triangles. When the included angle is 90 degrees, the formula reduces to the Pythagorean theorem since cos(90) = 0.
Law of Sines
Use this when you know one side and two angles (AAS or ASA condition). Since the angles of a triangle sum to 180 degrees, knowing two angles lets you find the third, then use the ratio a/sin(A) = b/sin(B) to find any unknown side.
Tips for Accurate Calculations
- Verify that your triangle is valid (the sum of any two sides must exceed the third).
- For the Pythagorean theorem, ensure the triangle is a right triangle.
- Angles in a triangle always sum to exactly 180 degrees.
- Convert angles to radians when using trigonometric functions in manual calculations.
- Check your answer by verifying it satisfies the triangle inequality.