Find the Side of a Triangle Given Two Sides Calculator (Law of Cosines)
Easily calculate the third side of any triangle using our find the side of a triangle given two sides calculator. Input two sides and the angle between them to get the missing side length using the Law of Cosines.
Third Side of Triangle Calculator
Side ‘c’ vs. Angle ‘C’ (for current ‘a’ and ‘b’)
What is the Third Side of Triangle Calculator?
The find the side of a triangle given two sides calculator, often referred to as a Law of Cosines calculator or a third side of triangle calculator, is a tool used to determine the length of the third side of any triangle when you know the lengths of the other two sides and the angle between them. It’s based on the Law of Cosines, a fundamental theorem in trigonometry that relates the lengths of the sides of a triangle to the cosine of one of its angles.
This calculator is invaluable for students, engineers, architects, and anyone working with geometry or real-world problems involving triangular shapes or distances. If you only know two sides and not the angle between them, you cannot uniquely determine the third side unless it’s a right-angled triangle (where the angle between the two legs is 90 degrees), in which case you might use a Pythagorean theorem calculator.
Common misconceptions include thinking you can find the third side with just two sides (you need an angle too, or knowledge it’s a right triangle and which sides you have) or that it only applies to right-angled triangles (the Law of Cosines applies to *all* triangles).
Third Side of Triangle (Law of Cosines) Formula and Mathematical Explanation
The third side of triangle calculator uses the Law of Cosines. If you have a triangle with sides labeled ‘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 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))
Note: The angle ‘C’ must be converted from degrees to radians for the cosine function in most programming languages (radians = degrees * π / 180).
Variables Table
| Variable | Meaning | Unit | Typical Range |
|---|---|---|---|
| a | Length of the first known side | Length units (e.g., m, cm, ft) | > 0 |
| b | Length of the second known side | Length units (e.g., m, cm, ft) | > 0 |
| C | Angle between sides a and b | Degrees | 0 < C < 180 |
| c | Length of the third side (opposite angle C) | Length units (e.g., m, cm, ft) | > 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 100 meters and 120 meters. The angle between these two sides is measured as 75 degrees. To find the length of the third side:
- a = 100 m
- b = 120 m
- C = 75 degrees
- c = √(100² + 120² – 2 * 100 * 120 * cos(75°))
- c ≈ √(10000 + 14400 – 24000 * 0.2588) ≈ √(24400 – 6211.2) ≈ √18188.8 ≈ 134.87 meters
The third side is approximately 134.87 meters long. Our find the side of a triangle given two sides calculator can quickly give this result.
Example 2: Navigation
A boat travels 15 km on one bearing, then changes direction and travels 20 km on another bearing. If the angle between the two paths was 110 degrees, how far is the boat from its starting point?
- a = 15 km
- b = 20 km
- C = 110 degrees
- c = √(15² + 20² – 2 * 15 * 20 * cos(110°))
- c ≈ √(225 + 400 – 600 * (-0.3420)) ≈ √(625 + 205.2) ≈ √830.2 ≈ 28.81 km
The boat is approximately 28.81 km from its starting point. This is easily found using the third side of triangle calculator.
How to Use This Find the Side of a Triangle Given Two Sides Calculator
- Enter Side ‘a’: Input the length of the first known side into the “Length of Side ‘a'” field.
- Enter Side ‘b’: Input the length of the second 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.
- Calculate: The calculator automatically updates the results as you type. You can also click “Calculate”.
- Read Results: The primary result is the length of side ‘c’. Intermediate calculations are also shown.
- Use Chart: The chart below the calculator visualizes how side ‘c’ changes with angle ‘C’ for the entered ‘a’ and ‘b’.
- Reset: Click “Reset” to clear the fields to default values.
- Copy: Click “Copy Results” to copy the main result and intermediate values.
The third side of triangle calculator provides the length of the missing side ‘c’, helping in various geometric and real-world measurements.
Key Factors That Affect Third Side Calculation Results
- Length of Side ‘a’: The longer side ‘a’ is, the longer side ‘c’ can be, depending on ‘b’ and ‘C’.
- Length of Side ‘b’: Similarly, the length of side ‘b’ directly influences the potential length of ‘c’.
- Angle ‘C’: This is crucial.
- If C is small (near 0°), c is close to |a-b|.
- If C is 90° (right angle), c = √(a²+b²) (Pythagorean theorem).
- If C is large (near 180°), c is close to a+b.
- Units: Ensure sides ‘a’ and ‘b’ are in the same units. The result ‘c’ will be in those same units. The calculator itself is unit-agnostic.
- Accuracy of Inputs: The precision of the calculated side ‘c’ depends directly on the precision of the input lengths and angle.
- Angle Measurement: Ensure the angle is correctly measured between the two known sides.
Our find the side of a triangle given two sides calculator accurately processes these inputs.
Frequently Asked Questions (FAQ)
A: You cannot uniquely determine the third side of a general triangle with only two sides. You need more information, like the angle between them, another angle, or if it’s a right-angled triangle (in which case you might use our Pythagorean theorem calculator).
A: Yes, the Law of Cosines, which this third side of triangle calculator uses, applies to all triangles (acute, obtuse, right-angled).
A: If C = 90 degrees, cos(90°) = 0, and the formula simplifies to c² = a² + b², which is the Pythagorean theorem. Our calculator handles this correctly.
A: You can use any unit of length (cm, m, inches, feet, etc.), but make sure both sides ‘a’ and ‘b’ are in the SAME unit. The result ‘c’ will be in that same unit.
A: In a triangle, any angle must be between 0 and 180 degrees (exclusive of 0 and 180). Our calculator accepts inputs between 0 and 180.
A: This calculator finds a side. To find angles given three sides, you would use a rearranged Law of Cosines, or our angle of triangle calculator.
A: The calculator uses standard mathematical formulas and is as accurate as the input values you provide.
A: The Law of Sines relates sides to the sines of opposite angles (a/sin(A) = b/sin(B) = c/sin(C)). It’s useful when you know a side and its opposite angle, plus one other side or angle. See our Law of Sines calculator.
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