Find Missing Side of Triangle Calculator
Enter the known values for your triangle below. You must provide at least three values (sides or angles) and leave exactly one side blank to find its length. Angles should be entered in degrees.
Result:
Understanding Triangles and Finding Missing Sides
A triangle is a fundamental geometric shape with three sides and three angles. The sum of the interior angles of any triangle always equals 180 degrees. Finding a missing side of a triangle is a common problem in geometry, trigonometry, engineering, and various scientific fields. This calculator helps you solve for an unknown side given sufficient information about the other sides and angles.
Triangle Notation
For clarity, we typically label the sides of a triangle as 'a', 'b', and 'c'. The angles opposite these sides are labeled 'A', 'B', and 'C' respectively. So, Angle A is opposite Side a, Angle B is opposite Side b, and Angle C is opposite Side c.
Key Principles for Finding Missing Sides
1. Pythagorean Theorem (for Right-Angled Triangles)
The Pythagorean Theorem applies exclusively to right-angled triangles (triangles with one 90-degree angle). If 'c' is the hypotenuse (the side opposite the right angle) and 'a' and 'b' are the other two legs, the theorem states:
a² + b² = c²
If you know any two sides of a right triangle, you can find the third. For example, if you know the two legs 'a' and 'b', you can find the hypotenuse 'c' using c = √(a² + b²). If you know the hypotenuse 'c' and one leg 'a', you can find the other leg 'b' using b = √(c² - a²).
2. Law of Cosines (SAS or SSS)
The Law of Cosines is a generalization of the Pythagorean Theorem and can be used for any triangle. It's particularly useful when you know:
- Two sides and the included angle (SAS – Side-Angle-Side): To find the third side.
- All three sides (SSS – Side-Side-Side): To find any angle (though our calculator focuses on sides).
The formulas are:
c² = a² + b² - 2ab * cos(C)a² = b² + c² - 2bc * cos(A)b² = a² + c² - 2ac * cos(B)
Where A, B, C are angles in degrees (converted to radians for calculation).
3. Law of Sines (AAS, ASA, or SSA)
The Law of Sines relates the sides of a triangle to the sines of its opposite angles. It's useful when you know:
- Two angles and any side (AAS – Angle-Angle-Side or ASA – Angle-Side-Angle): To find another side.
- Two sides and a non-included angle (SSA – Side-Side-Angle): This is the "ambiguous case" as it can sometimes yield two possible triangles, one or no triangle. Our calculator will provide one valid solution if it exists.
The formulas are:
a / sin(A) = b / sin(B) = c / sin(C)
From this, you can derive formulas to find a missing side, for example:
b = (a * sin(B)) / sin(A)c = (a * sin(C)) / sin(A)
Again, angles A, B, C are in degrees (converted to radians for calculation).
How to Use This Calculator
- Identify Known Values: Look at your triangle problem and determine which sides (a, b, c) and angles (A, B, C) you already know.
- Enter Values: Input these known values into the corresponding fields in the calculator. Leave the field for the side you want to find blank.
- Ensure Sufficient Information: You must provide at least three pieces of information (sides or angles) for the calculator to solve the triangle. Crucially, you must leave exactly one side blank.
- Click "Calculate": The calculator will automatically determine which formula to use (Pythagorean, Law of Cosines, or Law of Sines) based on your inputs and display the length of the missing side.
Examples
Example 1: Right-Angled Triangle (Pythagorean Theorem)
You have a right-angled triangle where one leg (Side a) is 6 units and the other leg (Side b) is 8 units. You want to find the hypotenuse (Side c).
- Input Side a: 6
- Input Side b: 8
- Input Angle C: 90 (or Angle A/B if it's the right angle)
- Leave Side c blank.
- Result: Side c = 10
Example 2: General Triangle (Law of Cosines – SAS)
You have a triangle where Side a is 7 units, Side b is 10 units, and the included Angle C is 60 degrees. You want to find Side c.
- Input Side a: 7
- Input Side b: 10
- Input Angle C: 60
- Leave Side c blank.
- Result: Side c ≈ 8.88
Example 3: General Triangle (Law of Sines – AAS)
You have a triangle where Angle A is 45 degrees, Angle B is 75 degrees, and Side a (opposite Angle A) is 12 units. You want to find Side b.
- Input Angle A: 45
- Input Angle B: 75
- Input Side a: 12
- Leave Side b blank.
- Result: Side b ≈ 16.39
This calculator simplifies complex trigonometric calculations, allowing you to quickly find missing side lengths for various triangle configurations.