Find the Third Side of a Triangle Calculator
Enter the lengths of two sides (a and b) and the angle (C) between them to find the length of the third side (c) using the Law of Cosines.
Results:
Visual representation of side lengths a, b, and c.
What is the Find the Third Side of a Triangle Calculator?
The "find the third side of a triangle calculator" is a tool that helps you determine the length of the unknown third side of a triangle when you know the lengths of the other two sides and the angle between them (the SAS – Side-Angle-Side case). This calculation is most commonly performed using the Law of Cosines, a fundamental theorem in trigonometry.
This calculator is useful for students, engineers, architects, and anyone dealing with geometric problems involving triangles where direct measurement of a side is difficult or impossible. It allows for quick and accurate calculation of the missing side length. Many people mistakenly think they always need a right-angled triangle to find sides, but the Law of Cosines (and our find the third side of a triangle calculator) works for any triangle.
Find the Third Side of a Triangle Calculator: Formula and Mathematical Explanation
To find the third side of a triangle when two sides and the included angle are known, we use the Law of Cosines. The formula is:
c² = a² + b² – 2ab cos(C)
Where:
- 'a' and 'b' are the lengths of the two known sides.
- 'C' is the angle (in radians or degrees) between sides 'a' and 'b'.
- 'c' is the length of the third side, opposite angle C.
The steps to find 'c' are:
- If the angle C is given in degrees, convert it to radians: Radians = Degrees × (π / 180).
- Calculate the cosine of angle C (in radians).
- Plug the values of a, b, and cos(C) into the formula c² = a² + b² – 2ab cos(C).
- Calculate c²
- Take the square root of c² to find the length of side c.
Variables Table
| Variable | Meaning | Unit | Typical Range |
|---|---|---|---|
| a | Length of the first known side | Length (e.g., m, cm, units) | > 0 |
| b | Length of the second known side | Length (e.g., m, cm, units) | > 0 |
| C | Angle between sides a and b | Degrees (or Radians for calculation) | 0° < C < 180° |
| c | Length of the third side (unknown) | Length (e.g., m, cm, units) | > 0 |
Our find the third side of a triangle calculator automates these steps for you.
Practical Examples (Real-World Use Cases)
Example 1: Surveying Land
A surveyor measures two sides of a triangular plot of land as 120 meters and 150 meters. The angle between these two sides is measured to be 75 degrees. To find the length of the third side, they use the find the third side of a triangle calculator (or the Law of Cosines).
- a = 120 m
- b = 150 m
- C = 75°
c² = 120² + 150² – 2 * 120 * 150 * cos(75°) ≈ 14400 + 22500 – 36000 * 0.2588 ≈ 36900 – 9316.8 = 27583.2
c ≈ √27583.2 ≈ 166.08 meters. The third side is approximately 166.08 meters long.
Example 2: Navigation
A boat travels 30 nautical miles, then changes direction and travels 40 nautical miles. The angle between its two legs of the journey is 110 degrees. How far is the boat from its starting point?
- a = 30 nm
- b = 40 nm
- C = 110°
c² = 30² + 40² – 2 * 30 * 40 * cos(110°) ≈ 900 + 1600 – 2400 * (-0.3420) ≈ 2500 + 820.8 = 3320.8
c ≈ √3320.8 ≈ 57.63 nautical miles. The boat is about 57.63 nautical miles from its start.
Using a law of sines calculator might be useful if different information was known.
How to Use This Find the Third Side of a Triangle Calculator
- Enter Side a: Input the length of one of the known sides into the "Side a" field.
- Enter Side b: Input the length of the other known side into the "Side b" field.
- Enter Angle C: Input the angle (in degrees) between sides a and b into the "Angle C" field. Ensure the angle is between 0 and 180 degrees.
- Calculate: Click the "Calculate Side c" button or simply change the input values. The calculator will automatically update the results.
- Read Results: The primary result, the length of side c, will be displayed prominently. Intermediate values like angle C in radians and c² are also shown.
- Reset: Click "Reset" to clear inputs and results to default values.
- Copy: Click "Copy Results" to copy the main result and inputs to your clipboard.
The find the third side of a triangle calculator gives you a quick answer based on the Law of Cosines.
Key Factors That Affect the Third Side Results
- Length of Side a: The longer side 'a' is, the longer side 'c' is likely to be, especially if angle C is large.
- Length of Side b: Similar to side 'a', the length of side 'b' directly influences the length of 'c'.
- Magnitude of Angle C: This is crucial.
- If C is close to 0°, c is close to |a – b|.
- If C is 90° (right angle), c² = a² + b² (Pythagorean theorem, a special case of Law of Cosines where cos(90°)=0). See our Pythagorean theorem calculator.
- If C is close to 180°, c is close to a + b.
- Units Used: Ensure the units for side a and side b are the same. The unit for side c will be the same as a and b.
- Accuracy of Input Values: Small errors in measuring 'a', 'b', or 'C' can lead to inaccuracies in the calculated value of 'c', especially when the angle is very small or very close to 180 degrees.
- Angle Units: Our calculator assumes Angle C is in degrees. Make sure your input matches this.
Understanding these factors helps in interpreting the results from the find the third side of a triangle calculator and when using a triangle solver for more complex problems.
Frequently Asked Questions (FAQ)
What if I know two angles and one side, or three sides?
If you know two angles and one side (ASA or AAS), you'd use the Law of Sines to find other sides. If you know three sides (SSS), you can use the Law of Cosines to find the angles, or check if the triangle is valid (sum of two sides > third side). This find the third side of a triangle calculator is specifically for the SAS (Side-Angle-Side) case.
Can I use this find the third side of a triangle calculator for a right-angled triangle?
Yes. If angle C is 90 degrees, cos(90°) = 0, and the formula c² = a² + b² – 2ab*0 simplifies to c² = a² + b², which is the Pythagorean theorem. You can also use our specific right-triangle calculator.
What happens if the angle is 0 or 180 degrees?
A triangle cannot have an angle of 0 or 180 degrees. If the angle is very close to 0, the three points are nearly collinear with c ≈ |a-b|. If close to 180, they are nearly collinear with c ≈ a+b.
Are there other ways to find the third side?
Besides the Law of Cosines (used by this find the third side of a triangle calculator), if you have different information (like two angles and a side), you would use the Law of Sines. For right triangles, Pythagorean theorem or basic trig ratios (SOH CAH TOA) can be used.
Why is it called the Law of Cosines?
Because it relates the lengths of the sides of a triangle to the cosine of one of its angles.
What units should I use?
You can use any unit of length (meters, feet, cm, inches, etc.) for sides 'a' and 'b', as long as they are consistent. The result for side 'c' will be in the same unit. Angle C must be in degrees for this calculator.
Does the calculator check if the inputs form a valid triangle?
For the SAS case, as long as the sides are positive and the angle is between 0 and 180 (exclusive), a unique triangle is formed. The find the third side of a triangle calculator will give a valid side 'c'.
Where else is the Law of Cosines used?
It's used in physics (vector addition), engineering, navigation, astronomy, and many other fields where distances and angles in non-right triangles are involved.
Related Tools and Internal Resources
- Triangle Area Calculator: Calculate the area of a triangle using various formulas.
- Pythagorean Theorem Calculator: For right-angled triangles specifically.
- Right Triangle Calculator: Solves various aspects of a right triangle.
- Law of Sines Calculator: Used when you know AAS or ASA to find sides.
- Geometry Calculators: A collection of calculators for various geometric shapes.
- Math Calculators: Our main hub for mathematical calculators.