Third Side of a Triangle Calculator
Easily find the length of the third side of a triangle using our calculator based on the Law of Cosines, given two sides and the angle between them.
Calculate the Third Side
Triangle Visualization
Visual representation of the triangle with the calculated third side ‘c’.
Example Calculations
| Side ‘a’ | Side ‘b’ | Angle ‘C’ (Degrees) | Side ‘c’ (Calculated) |
|---|---|---|---|
| 3 | 4 | 30 | 2.05 |
| 3 | 4 | 60 | 3.61 |
| 3 | 4 | 90 | 5.00 |
| 5 | 12 | 90 | 13.00 |
| 5 | 5 | 120 | 8.66 |
Table showing the calculated third side ‘c’ for different input values of ‘a’, ‘b’, and Angle ‘C’.
What is a Third Side of a Triangle Calculator?
A third side of a triangle calculator is a tool used to determine the length of the unknown side of a triangle when you know the lengths of the other two sides and the angle between them. This is typically done using the Law of Cosines. It’s particularly useful in geometry, trigonometry, engineering, and physics when you can’t directly measure a side but have information about the other two sides and the included angle.
Anyone dealing with triangles, from students learning trigonometry to professionals in fields like surveying, architecture, or navigation, can benefit from using a third side of a triangle calculator. It saves time and reduces the chance of manual calculation errors.
A common misconception is that you always need a right-angled triangle to find a missing side (using the Pythagorean theorem). However, the Law of Cosines, which this third side of a triangle calculator uses, applies to ANY triangle, not just right-angled ones.
Third Side of a Triangle Calculator Formula and Mathematical Explanation
The most common formula used by a third side of a triangle calculator when two sides and the included angle are known is the Law of Cosines. If we have a triangle with sides ‘a’, ‘b’, and ‘c’, and the angles opposite these sides are A, B, and C respectively, the Law of Cosines states:
c² = a² + b² – 2ab cos(C)
Where:
- ‘a’ and ‘b’ are the lengths of the 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 (the side we want to find).
- cos(C) is the cosine of angle C.
To find ‘c’, we take the square root of the result:
c = √(a² + b² – 2ab cos(C))
The calculator first converts the angle C from degrees to radians (since most trigonometric functions in programming use radians), then calculates cos(C), plugs the values into the formula, and finally finds the square root to get the length of side ‘c’. Our third side of a triangle calculator performs these steps automatically.
Variables Table
| Variable | Meaning | Unit | Typical Range |
|---|---|---|---|
| a | Length of the first known side | Length units (e.g., cm, m, inches) | > 0 |
| b | Length of the second known side | Length units (e.g., cm, m, inches) | > 0 |
| C | Angle between sides a and b | Degrees | 0 < C < 180 |
| c | Length of the third side (unknown) | Length units (e.g., cm, m, inches) | Depends on a, b, and C |
| cos(C) | Cosine of angle C | Dimensionless | -1 to 1 |
Practical Examples (Real-World Use Cases)
Example 1: Surveying a Plot of Land
A surveyor needs to find the length of one side of a triangular plot of land. They measure two sides as 50 meters and 70 meters, and the angle between these two sides is 60 degrees.
- Side a = 50 m
- Side b = 70 m
- Angle C = 60 degrees
Using the third side of a triangle calculator (or the formula c = √(50² + 70² – 2 * 50 * 70 * cos(60°))):
c = √(2500 + 4900 – 7000 * 0.5) = √(7400 – 3500) = √3900 ≈ 62.45 meters.
The length of the third side is approximately 62.45 meters.
Example 2: Navigation
A boat travels 5 km due east, then changes direction and travels 8 km on a bearing such that the angle between its initial path and the new path is 110 degrees. How far is the boat from its starting point?
- Side a = 5 km
- Side b = 8 km
- Angle C = 110 degrees
Using the third side of a triangle calculator (c = √(5² + 8² – 2 * 5 * 8 * cos(110°))):
cos(110°) ≈ -0.342
c = √(25 + 64 – 80 * (-0.342)) = √(89 + 27.36) = √116.36 ≈ 10.79 km.
The boat is approximately 10.79 km from its starting point.
How to Use This Third Side of a Triangle 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 between sides ‘a’ and ‘b’ in degrees into the “Angle ‘C'” field. Ensure the angle is between 0 and 180 degrees.
- Calculate: Click the “Calculate” button or simply change any input value. The third side of a triangle calculator will automatically update the results.
- Read the Results: The primary result is the length of the third side ‘c’, displayed prominently. Intermediate values like the angle in radians and c² are also shown.
- Visualize: The SVG diagram provides a visual representation of the triangle based on your inputs.
- Reset: Click “Reset” to return to the default values.
- Copy: Click “Copy Results” to copy the calculated values to your clipboard.
This third side of a triangle calculator is a straightforward tool for quickly finding the missing side.
Key Factors That Affect Third Side of a Triangle Calculator Results
- Length of Side ‘a’: The longer side ‘a’ is, the longer side ‘c’ will generally be, especially if the angle is large.
- Length of Side ‘b’: Similar to side ‘a’, the length of side ‘b’ directly influences the length of side ‘c’.
- Angle ‘C’: This is crucial.
- As Angle ‘C’ approaches 0 degrees, side ‘c’ approaches the absolute difference between ‘a’ and ‘b’ (|a – b|).
- As Angle ‘C’ approaches 180 degrees, side ‘c’ approaches the sum of ‘a’ and ‘b’ (a + b).
- When Angle ‘C’ is 90 degrees, the Law of Cosines reduces to the Pythagorean theorem (c² = a² + b²), as cos(90°) = 0.
- 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. The third side of a triangle calculator doesn’t convert units, it just performs the calculation.
- Accuracy of Input: The precision of the calculated side ‘c’ depends on the precision of the input values for ‘a’, ‘b’, and angle ‘C’.
- Angle Measurement: Make sure the angle is entered in degrees, as required by this third side of a triangle calculator. The internal conversion to radians is handled automatically.
Frequently Asked Questions (FAQ)
1. What formula does the third side of a triangle calculator use?
It primarily uses the Law of Cosines: c² = a² + b² – 2ab cos(C), to find the third side ‘c’ given sides ‘a’, ‘b’, and the included angle ‘C’.
2. Can I use this calculator for a right-angled triangle?
Yes. If you enter 90 degrees for Angle ‘C’, the term -2ab cos(90°) becomes zero, and the formula simplifies to c² = a² + b², which is the Pythagorean theorem. You might find our Pythagorean theorem calculator more direct for right triangles.
3. What if I know two angles and one side, or three sides?
If you know three sides, you can use the Law of Cosines to find angles. If you know two angles and a side, or one angle and two sides (but not the included angle), you might need the Law of Sines. Check our Sine Rule calculator or Cosine Rule calculator for these cases.
4. What units should I use for the sides?
You can use any unit of length (meters, feet, cm, inches, etc.) for sides ‘a’ and ‘b’, as long as you use the SAME unit for both. The result for side ‘c’ will be in that same unit. The third side of a triangle calculator is unit-agnostic in that sense.
5. What is the valid range for the angle C?
For a non-degenerate triangle, the angle C must be greater than 0 degrees and less than 180 degrees (0 < C < 180).
6. Why does the calculator show intermediate results?
Intermediate results like the angle in radians and c² help you understand the steps involved in the calculation and can be useful for verification or further calculations.
7. How accurate is this third side of a triangle calculator?
The calculator uses standard mathematical functions and provides high precision based on your input. The accuracy of the final result depends on the accuracy of your input values.
8. Can I find angles using this calculator?
No, this specific third side of a triangle calculator is designed to find the length of the third side given two sides and the included angle. To find angles, you would rearrange the Law of Cosines or use the Law of Sines. Consider our triangle angle calculator.
Related Tools and Internal Resources
- Triangle Area Calculator: Calculate the area of a triangle using various formulas.
- Pythagorean Theorem Calculator: Specifically for finding sides of right-angled triangles.
- Right Triangle Calculator: Solves for sides and angles of right triangles.
- Sine Rule Calculator: Useful when you know two sides and a non-included angle, or two angles and a side.
- Cosine Rule Calculator: Another name for the Law of Cosines, can find sides or angles.
- Triangle Angle Calculator: Find the angles of a triangle given side lengths.