Triangle Length Calculator
Calculate Side Length 'c' using Law of Cosines
Enter the lengths of two sides (a and b) and the angle (C) between them to find the length of the third side (c).
What is a Triangle Length Calculator?
A triangle length calculator is a tool used to determine the length of one or more sides of a triangle based on other known information, such as the lengths of other sides and the measures of angles. This specific calculator focuses on using the Law of Cosines to find the length of the third side (c) when the lengths of two sides (a and b) and the included angle (C) are known.
It's particularly useful for non-right-angled triangles where the Pythagorean theorem doesn't directly apply to find all sides. Anyone working with geometry, trigonometry, engineering, construction, or even navigation might use a triangle length calculator. A common misconception is that you always need a right angle to find triangle sides, but tools like the Law of Cosines and Law of Sines calculators work for any triangle.
Triangle Length Formula and Mathematical Explanation (Law of Cosines)
To find the length of a side of a triangle when you know the lengths of the other two sides and the angle between them, we use the Law of Cosines. The formula is:
c² = a² + b² – 2ab cos(C)
Where:
- 'a' and 'b' are the lengths of two known sides.
- 'C' is the measure of the angle between sides 'a' and 'b'.
- 'c' is the length of the side opposite angle C, which we want to find.
To find 'c', we take the square root of the result:
c = √(a² + b² – 2ab cos(C))
When using the formula, ensure the angle 'C' is converted to radians if your calculator or programming language's cosine function expects radians (Angle in Radians = Angle in Degrees * π / 180).
Variables Table
| Variable | Meaning | Unit | Typical Range |
|---|---|---|---|
| a | Length of side 'a' | Length units (e.g., cm, m, inches) | > 0 |
| b | Length of side 'b' | Length units (e.g., cm, m, inches) | > 0 |
| C | Angle between sides 'a' and 'b' | Degrees | 0 < C < 180 |
| c | Length of side 'c' (opposite angle C) | Length units (e.g., cm, m, inches) | > 0 |
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, with the angle between these two sides being 60 degrees. They need to find the length of the third side.
- Side a = 120 m
- Side b = 150 m
- Angle C = 60 degrees
Using the triangle length calculator (or Law of Cosines): c = √(120² + 150² – 2 * 120 * 150 * cos(60°)) c = √(14400 + 22500 – 36000 * 0.5) c = √(36900 – 18000) = √18900 ≈ 137.48 meters The third side is approximately 137.48 meters.
Example 2: Navigation
A boat travels 10 km on one bearing, then changes direction and travels 8 km on another bearing. If the angle between the two paths was 110 degrees, how far is the boat from its starting point in a straight line?
- Side a = 10 km
- Side b = 8 km
- Angle C = 110 degrees
Using the triangle length calculator: c = √(10² + 8² – 2 * 10 * 8 * cos(110°)) c = √(100 + 64 – 160 * (-0.3420)) c = √(164 + 54.72) = √218.72 ≈ 14.79 km The boat is approximately 14.79 km from its starting point.
How to Use This Triangle Length Calculator
- Enter Side a: Input the length of one of the known sides into the "Length of Side a" field.
- Enter Side b: Input the length of the other known side into the "Length of 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 (exclusive of 0 and 180 for a valid triangle side calculation using this method).
- Calculate: The calculator will automatically update the results as you type, or you can click "Calculate".
- View Results: The "Length of Side c" will be displayed prominently, along with intermediate values like the angle in radians and c².
- Reset: Click "Reset" to clear the fields and start over with default values.
- Copy Results: Click "Copy Results" to copy the main result and intermediate values to your clipboard.
The results from the triangle length calculator give you the direct length of the unknown side. The chart visually compares the lengths of all three sides.
Key Factors That Affect Triangle Length Results
- Accuracy of Side Measurements: Small errors in measuring sides 'a' and 'b' can lead to inaccuracies in the calculated length of 'c'. Ensure your input measurements are as precise as possible.
- Accuracy of Angle Measurement: The angle 'C' is crucial. An error in measuring the angle will significantly impact the result, especially as the angle approaches 0 or 180 degrees.
- Units Used: Ensure that the units for side 'a' and side 'b' are the same. The calculated side 'c' will be in the same units.
- Angle Units: This calculator expects the angle in degrees. If your angle is in radians, convert it to degrees first (Degrees = Radians * 180 / π).
- Rounding: The final result and intermediate calculations might involve rounding, which can introduce very small differences depending on the precision used.
- Calculator Precision: The precision of the cosine function and square root function used in the calculation can affect the final digits of the result.
Using a reliable triangle length calculator and providing accurate inputs are key to getting a correct result.
Frequently Asked Questions (FAQ)
- 1. What is the Law of Cosines?
- The Law of Cosines is a formula relating the lengths of the sides of a triangle to the cosine of one of its angles. It's used in our triangle length calculator to find a side when two sides and the included angle are known (c² = a² + b² – 2ab cos(C)).
- 2. Can I use this calculator for a right-angled triangle?
- Yes, you can. If angle C is 90 degrees, cos(90°) = 0, and the formula becomes c² = a² + b², which is the Pythagorean theorem. However, for right triangles, a Pythagorean theorem calculator might be more direct if you know two sides and want the third.
- 3. What if I know two angles and one side?
- If you know two angles and one side, you should use the Law of Sines to find the other sides. Check out our Law of Sines calculator.
- 4. What if I know all three sides and want to find an angle?
- You can rearrange the Law of Cosines to solve for an angle: cos(C) = (a² + b² – c²) / 2ab. You'd then use an angle calculator or inverse cosine function.
- 5. Why is the angle restricted to between 0 and 180 degrees?
- The internal angles of a triangle always sum to 180 degrees, and each individual angle must be greater than 0 and less than 180 degrees for a valid triangle.
- 6. What units should I use for the sides?
- You can use any unit of length (cm, meters, inches, feet, etc.) as long as you are consistent for both side 'a' and side 'b'. The result for side 'c' will be in the same unit.
- 7. How accurate is this triangle length calculator?
- The calculator is as accurate as the input values you provide and the mathematical precision of JavaScript's Math functions. Ensure your input measurements are correct.
- 8. Can I find the area using these inputs?
- Yes, once you know two sides and the included angle, the area of the triangle is (1/2) * a * b * sin(C). See our triangle area calculator.