Find the Length of the Third Side Calculator (Law of Cosines)
Easily calculate the length of the third side of any triangle (side 'c') given the lengths of two sides ('a' and 'b') and the angle ('C') between them using our find the length of the third side calculator.
Triangle Side Calculator
Understanding the Calculation
Our find the length of the third side calculator uses the Law of Cosines to determine the length of the unknown side 'c' when you know the lengths of the other two sides ('a' and 'b') and the angle 'C' opposite to side 'c'.
Chart: The graph illustrates how the length of side 'c' changes as angle 'C' varies from 0 to 180 degrees, keeping side 'a' and 'b' constant at their current input values. The lines for a+b and |a-b| show the theoretical maximum and minimum values for 'c'.
What is a Find the Length of the Third Side Calculator?
A "find the length of the third side calculator" is a tool used in trigonometry to determine the length of one side of a triangle when the lengths of the other two sides and the angle between them are known. It most commonly employs the Law of Cosines, a fundamental theorem relating the lengths of the sides of a triangle to the cosine of one of its angles. This calculator is particularly useful for non-right-angled triangles, where the Pythagorean theorem does not directly apply to find any side.
Anyone dealing with geometric problems, such as surveyors, engineers, architects, navigators, and students studying trigonometry, can benefit from using a find the length of the third side calculator. It provides a quick and accurate way to solve for an unknown side without manual calculations.
A common misconception is that you can always use the Pythagorean theorem (a² + b² = c²) to find the third side. However, this is only true for right-angled triangles, where 'c' is the hypotenuse. For any other triangle, the find the length of the third side calculator uses the more general Law of Cosines.
Find the Length of the Third Side Formula and Mathematical Explanation
The primary formula used by our find the length of the third side calculator is the Law of Cosines:
c² = a² + b² – 2ab * cos(C)
Where:
- 'c' is the length of the side opposite angle 'C' (the side we want to find).
- 'a' and 'b' are the lengths of the other two sides.
- 'C' is the angle between sides 'a' and 'b', opposite side 'c'.
- cos(C) is the cosine of angle 'C'.
To find 'c', we take the square root of the right side of the equation:
c = √(a² + b² – 2ab * cos(C))
The find the length of the third side calculator performs these steps automatically.
Variables Table
| Variable | Meaning | Unit | Typical Range |
|---|---|---|---|
| a | Length of the first known side | Units (e.g., m, cm, ft, inches) | > 0 |
| b | Length of the second known side | Units (e.g., m, cm, ft, inches) | > 0 |
| C | Angle between sides 'a' and 'b' | Degrees | 0° < C < 180° |
| c | Length of the third side (to be calculated) | Same units as 'a' and 'b' | > 0 |
Table: Variables involved in the Law of Cosines for finding the third side.
Practical Examples (Real-World Use Cases)
Example 1: Land Surveying
A surveyor measures two sides of a triangular piece 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 of the land, they use the find the length of the third side calculator:
- Side a = 120 m
- Side b = 150 m
- Angle C = 75°
Using the formula c = √(120² + 150² – 2 * 120 * 150 * cos(75°)), the calculator finds c ≈ √(14400 + 22500 – 36000 * 0.2588) ≈ √(36900 – 9316.8) ≈ √27583.2 ≈ 166.08 meters. The third side is approximately 166.08 meters long.
Example 2: Navigation
A boat travels 20 nautical miles from point A to point B, then changes direction by 110 degrees and travels 30 nautical miles to point C. To find the direct distance between point A and point C, we have a triangle where two sides are 20 and 30 nautical miles, and the angle *between* these paths at B is (180 – 110) = 70 degrees (if 110 was the exterior angle of turn, or 110 if it was the interior angle of the triangle at B – let's assume the angle within the triangle at B is 110 degrees, so the angle opposite AC is 110).
- Side a = 20 nautical miles
- Side b = 30 nautical miles
- Angle C = 110° (angle at B, opposite side AC)
Using the find the length of the third side calculator: c = √(20² + 30² – 2 * 20 * 30 * cos(110°)) = √(400 + 900 – 1200 * (-0.3420)) ≈ √(1300 + 410.4) ≈ √1710.4 ≈ 41.36 nautical miles. The direct distance is about 41.36 nautical miles.
How to Use This Find the Length of the Third Side 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. This is the angle opposite the side 'c' you want to find.
- Read Results: The calculator will instantly display the length of the third side 'c', along with intermediate calculations like a², b², 2ab, and cos(C).
- Reset: Click the "Reset" button to clear the inputs and results and start a new calculation with default values.
- Copy: Click "Copy Results" to copy the calculated side 'c' and intermediate values to your clipboard.
The find the length of the third side calculator provides an immediate result based on your inputs, allowing for quick checks and problem-solving.
Key Factors That Affect the Length of the Third Side
- Length of Side 'a': Increasing the length of side 'a' (while b and C are constant) will generally increase the length of side 'c'.
- Length of Side 'b': Similarly, increasing the length of side 'b' (while a and C are constant) will generally increase the length of side 'c'.
- Magnitude of Angle 'C': The angle C is crucial. As angle C increases from 0° towards 180°, the value of cos(C) decreases from 1 to -1.
- When C is small (near 0°), cos(C) is near 1, making -2ab*cos(C) a large negative number, so 'c' is small (c ≈ |a-b|).
- When C = 90°, cos(C) = 0, and c² = a² + b² (Pythagorean theorem).
- When C is large (near 180°), cos(C) is near -1, making -2ab*cos(C) a large positive number, so 'c' is large (c ≈ a+b).
- Units Used: Ensure that the units for sides 'a' and 'b' are consistent. The calculated side 'c' will be in the same units.
- Accuracy of Angle Measurement: Small errors in measuring angle C can lead to significant differences in the calculated length of 'c', especially when 'a' and 'b' are large or when C is close to 0 or 180 degrees.
- 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. The calculator assumes valid inputs that can form a triangle, but the result 'c' must satisfy |a-b| < c < a+b.
Understanding these factors helps in interpreting the results from the find the length of the third side calculator.
Frequently Asked Questions (FAQ)
When should I use the Law of Cosines instead of the Pythagorean theorem?
Use the Pythagorean theorem (a² + b² = c²) only for right-angled triangles to find the hypotenuse 'c' or one of the other sides if the hypotenuse is known. Use the Law of Cosines (and our find the length of the third side calculator) for ANY triangle, especially non-right-angled ones, when you know two sides and the included angle (SAS) or all three sides (SSS – to find an angle).
What if I know two angles and one side?
If you know two angles and one side, you first find the third angle (sum of angles = 180°) and then use the Law of Sines to find the other sides. This calculator is for when you know two sides and the included angle (SAS).
What units should I use for the side lengths?
You can use any units (meters, feet, cm, inches, etc.) for the side lengths, as long as you are consistent for both side 'a' and side 'b'. The resulting side 'c' will be in the same units.
Can the angle C be greater than 180 degrees?
No, in a triangle, any interior angle must be between 0 and 180 degrees (exclusive of 0 and 180).
What happens if I enter an angle of 90 degrees?
If you enter 90 degrees for angle C, cos(90°) = 0, and the Law of Cosines formula reduces to c² = a² + b², which is the Pythagorean theorem. The find the length of the third side calculator will give you the hypotenuse.
Can I use this find the length of the third side calculator for any triangle?
Yes, the Law of Cosines applies to all triangles, whether they are acute, obtuse, or right-angled, provided you have the lengths of two sides and the angle between them.
What if my input values don't form a valid triangle?
The formula itself doesn't prevent calculation, but if the angle is 0 or 180, or sides are zero or negative, it's not a standard triangle. The calculator validates for angle between 0 and 180 and positive sides.
How accurate is this find the length of the third side calculator?
The calculator uses standard mathematical formulas and is as accurate as the input values you provide. Ensure your side lengths and angle measurements are precise for an accurate result.
Related Tools and Internal Resources
- Right Triangle Calculator: Calculate sides and angles of a right triangle.
- Triangle Area Calculator: Find the area of a triangle using various methods.
- Law of Sines Calculator: Solve triangles when you know two angles and one side, or two sides and a non-included angle.
- Pythagorean Theorem Calculator: Specifically for right-angled triangles.
- Geometry Calculators: Explore more calculators related to geometric shapes.
- Angle Converter: Convert between degrees and radians.