Find the Third Length of a Triangle Calculator (Law of Cosines)
Use this calculator to find the length of the third side of a triangle when you know the lengths of the other two sides and the angle between them (SAS – Side-Angle-Side). This is based on the Law of Cosines.
What is the Find the Third Length of a Triangle Calculator?
The find the third length of a triangle calculator is a tool used in geometry and trigonometry to determine the length of an unknown side of a triangle when you have information about the other sides and/or angles. Most commonly, when you know two sides and the angle between them (the SAS case – Side-Angle-Side), this calculator uses the Law of Cosines to find the third side. It’s an essential tool for students, engineers, architects, and anyone dealing with geometric calculations.
This specific calculator focuses on the Law of Cosines, which relates the lengths of the sides of a triangle to the cosine of one of its angles. If you have a right-angled triangle, you might use the Pythagorean theorem instead, but the Law of Cosines works for any triangle.
Who should use it? Students learning trigonometry, engineers designing structures, architects planning buildings, or even DIY enthusiasts measuring spaces will find the find the third length of a triangle calculator incredibly useful.
A common misconception is that you always need all angles or that it only works for right triangles. The Law of Cosines, used by this find the third length of a triangle calculator, is powerful because it applies to any triangle, not just right-angled ones, provided you have the correct inputs (two sides and the included angle).
Find the Third Length of a Triangle Calculator Formula (Law of Cosines) and Mathematical Explanation
The primary formula used by the find the third length of a triangle calculator when given two sides (a and b) and the included angle (C) is the Law of Cosines:
c² = a² + b² – 2ab * cos(C)
To find the length of side ‘c’, we take the square root:
c = √(a² + b² – 2ab * cos(C))
Where:
- ‘a’ and ‘b’ are the lengths of the two known sides.
- ‘C’ is the angle (in radians for the cos function) between sides ‘a’ and ‘b’, and opposite to side ‘c’.
- ‘c’ is the length of the side we want to find.
Step-by-step derivation/explanation:
- Square the lengths of the known sides: a² and b².
- Add these squares: a² + b².
- Calculate the cosine of the angle C. Make sure the angle is in radians if your calculator/function requires it (degrees * π/180 = radians).
- Multiply 2, side a, side b, and cos(C): 2 * a * b * cos(C).
- Subtract the result from step 4 from the result from step 2: (a² + b²) – (2ab * cos(C)).
- Take the square root of the result from step 5 to find the length of side c.
Variables Table
| Variable | Meaning | Unit | Typical Range |
|---|---|---|---|
| a | Length of the first known side | Units (e.g., cm, m, inches) | > 0 |
| b | Length of the second known side | 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) | Units (e.g., cm, m, inches) | > 0 |
| cos(C) | Cosine of angle C | Dimensionless | -1 to 1 |
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 as 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 find the third length of a triangle calculator (or the Law of Cosines):
c = √(120² + 150² – 2 * 120 * 150 * cos(60°))
c = √(14400 + 22500 – 36000 * 0.5)
c = √(36900 – 18000)
c = √18900 ≈ 137.48 meters
The third side of the land is approximately 137.48 meters.
Example 2: Navigation
A ship sails 50 nautical miles on one bearing, then changes course and sails 70 nautical miles on another. If the angle between the two legs of the journey was 110 degrees, how far is the ship from its starting point in a straight line?
- Side a = 50 nm
- Side b = 70 nm
- Angle C = 110 degrees
Using the find the third length of a triangle calculator:
c = √(50² + 70² – 2 * 50 * 70 * cos(110°))
c = √(2500 + 4900 – 7000 * (-0.3420))
c = √(7400 + 2394.14)
c = √9794.14 ≈ 98.96 nautical miles
The ship is approximately 98.96 nautical miles from its starting point.
How to Use This Find the Third Length 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. This angle must be between 0 and 180 degrees non-inclusive.
- Calculate: Click the “Calculate Side c” button or simply change the input values (if auto-calculate is enabled, which it is on input change here).
- Read Results: The calculator will display the length of the third side ‘c’, along with intermediate values like the angle in radians.
- Interpret: The “Length of Side c” is the primary result you are looking for. The units will be the same as the units you used for sides a and b.
The find the third length of a triangle calculator provides a quick and accurate way to apply the Law of Cosines without manual calculation.
Key Factors That Affect the Third Length
The length of the third side ‘c’ is directly influenced by:
- Length of Side a: As side ‘a’ increases (with ‘b’ and ‘C’ constant), side ‘c’ will generally increase.
- Length of Side b: Similarly, as side ‘b’ increases (with ‘a’ and ‘C’ constant), side ‘c’ will generally increase.
- Magnitude of Angle C: This is crucial.
- If C is small (close to 0 degrees), cos(C) is close to 1, and ‘c’ will be close to |a – b|. The triangle is almost flat.
- If C is 90 degrees (right angle), cos(C) is 0, and c² = a² + b² (Pythagorean theorem).
- If C is large (close to 180 degrees), cos(C) is close to -1, and ‘c’ will be close to a + b. The triangle is again almost flat but with ‘a’ and ‘b’ nearly extending in opposite directions from the vertex of C before connecting via ‘c’.
- Units Used: The units of side ‘c’ will be the same as the units used for sides ‘a’ and ‘b’. Ensure consistency.
- Accuracy of Input: Small errors in the input values, especially the angle, can lead to noticeable differences in the calculated length of side ‘c’, particularly when sides are long or the angle is very small or very large.
- Triangle Inequality Theorem: For a valid triangle, the sum of the lengths of any two sides must be greater than the length of the third side (a + b > c, a + c > b, b + c > a). The Law of Cosines inherently respects this if 0 < C < 180.
Understanding these factors helps in interpreting the results from the find the third length of a triangle calculator.
Frequently Asked Questions (FAQ)
A: The Law of Cosines is a formula relating the lengths of the sides of a triangle to the cosine of one of its angles: c² = a² + b² – 2ab * cos(C). It’s used by this find the third length of a triangle calculator.
A: Use the Pythagorean theorem (a² + b² = c²) only for right-angled triangles to find a side. Use the Law of Cosines when you know two sides and the included angle (SAS) or all three sides (SSS) of ANY triangle, right-angled or not. Our find the third length of a triangle calculator uses it for the SAS case.
A: Not directly. If you know two angles and one side (ASA or AAS), you would typically use the Law of Sines first to find another side, then maybe the Law of Cosines or the fact that angles sum to 180 degrees. This calculator is specifically for the SAS case.
A: An angle of 0 or 180 degrees means the three vertices are collinear (form a straight line), not a triangle. The calculator expects an angle between 0 and 180 degrees (exclusive).
A: You can rearrange the Law of Cosines to solve for an angle: cos(C) = (a² + b² – c²) / 2ab. You’d need a different calculator or rearrange the formula yourself.
A: You must use the same units for both ‘a’ and ‘b’. The result for ‘c’ will be in those same units. The find the third length of a triangle calculator doesn’t convert units.
A: Yes, the Law of Cosines can be derived using the vector dot product. If sides a and b are represented by vectors, the dot product involves the cosine of the angle between them.
A: No, the length of a side of a triangle cannot be negative. The formula involves taking a square root of a value that will be non-negative for valid triangle inputs (0 < C < 180).
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