Find Side of a Triangle Not Right Calculator (Law of Cosines)
Easily calculate the third side of any non-right-angled triangle given two sides and the angle between them.
Triangle Side Calculator
Inputs and Result Summary
| Parameter | Value | Unit |
|---|---|---|
| Side a | 10 | units |
| Side b | 12 | units |
| Angle C | 60 | degrees |
| Side c | – | units |
Table showing the input values and the calculated length of side ‘c’.
Side ‘c’ Length vs. Angle ‘C’
Chart showing how the length of side ‘c’ changes as Angle ‘C’ varies from 1 to 179 degrees, with the current Side ‘a’ and Side ‘b’ values.
What is the ‘Find Side of a Triangle Not Right Calculator’?
The “find side of a triangle not right calculator” is a tool used to determine the length of an unknown side of a triangle that is not necessarily right-angled. When you know the lengths of two sides and the angle between them (SAS – Side-Angle-Side configuration), you can use the Law of Cosines to find the length of the third side. This calculator specifically applies the Law of Cosines.
Anyone needing to solve for triangle dimensions in fields like geometry, trigonometry, engineering, physics, surveying, or navigation can use this calculator. It’s particularly useful when you don’t have a right angle, making the Pythagorean theorem inapplicable.
A common misconception is that you can always use the Pythagorean theorem (a² + b² = c²) for any triangle; however, it only applies to right-angled triangles. For non-right triangles (oblique triangles), you need the Law of Cosines or the Law of Sines.
Find Side of a Triangle Not Right Calculator: Formula and Mathematical Explanation
To find the length of side ‘c’ when you know sides ‘a’, ‘b’, and the included angle ‘C’, we use the Law of Cosines:
c² = a² + b² – 2ab cos(C)
Therefore, side ‘c’ is:
c = √(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’, measured in degrees (which must be converted to radians for the `cos` function in calculations).
- ‘c’ is the length of the side opposite angle ‘C’, which we want to find.
Variables Table
| Variable | Meaning | Unit | Typical Range |
|---|---|---|---|
| a | Length of side ‘a’ | units (e.g., cm, m, ft) | > 0 |
| b | Length of side ‘b’ | units (e.g., cm, m, ft) | > 0 |
| C | Angle between sides ‘a’ and ‘b’ | degrees | 0 < C < 180 |
| c | Length of side ‘c’ (opposite angle C) | units (e.g., cm, m, ft) | > 0, dependent on a, b, C |
Practical Examples (Real-World Use Cases)
Let’s see how the find side of a triangle not right calculator works in practice.
Example 1: Surveying a Plot of Land
A surveyor measures two sides of a triangular plot of land as 150 meters and 200 meters. The angle between these two sides is measured as 75 degrees. What is the length of the third side?
- Side a = 150 m
- Side b = 200 m
- Angle C = 75 degrees
Using the formula c = √(150² + 200² – 2 * 150 * 200 * cos(75°)):
c = √(22500 + 40000 – 60000 * cos(75°)) ≈ √(62500 – 60000 * 0.2588) ≈ √(62500 – 15528) ≈ √46972 ≈ 216.73 meters. The third side is approximately 216.73 meters long.
Example 2: Navigation
A boat travels 30 nautical miles, then turns 110 degrees and travels 40 nautical miles. How far is the boat from its starting point in a straight line?
- Side a = 30 nm
- Side b = 40 nm
- Angle C = 110 degrees
Using the formula c = √(30² + 40² – 2 * 30 * 40 * cos(110°)):
c = √(900 + 1600 – 2400 * cos(110°)) ≈ √(2500 – 2400 * (-0.3420)) ≈ √(2500 + 820.8) ≈ √3320.8 ≈ 57.63 nautical miles. The boat is approximately 57.63 nautical miles from its start.
How to Use This Find Side of a Triangle Not Right 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.
- Calculate: The calculator will automatically update the results as you type, or you can click “Calculate Side ‘c'”.
- Read Results: The primary result is the length of side ‘c’. Intermediate values like angle C in radians are also shown.
- Analyze Chart: The chart shows how side ‘c’ would change if angle ‘C’ varied, keeping ‘a’ and ‘b’ constant.
- Reset: Click “Reset” to return to default values.
- Copy Results: Click “Copy Results” to copy the inputs and outputs to your clipboard.
The result for side ‘c’ will be in the same units as sides ‘a’ and ‘b’.
Key Factors That Affect Find Side of a Triangle Not Right Calculator Results
The length of the unknown side ‘c’ is primarily affected by:
- Length of Side ‘a’: The longer side ‘a’ is, the longer side ‘c’ is likely to be, especially if angle C is large.
- Length of Side ‘b’: Similar to side ‘a’, the length of side ‘b’ directly influences ‘c’.
- Magnitude of Angle ‘C’: This is crucial. If angle C is small (close to 0), ‘c’ will be close to |a-b|. If C is large (close to 180), ‘c’ will be close to a+b. If C is 90 degrees, it becomes a right triangle, and c² = a² + b² (as cos(90)=0).
- Units Used: Ensure ‘a’ and ‘b’ are in the same units; ‘c’ will be in those units.
- Accuracy of Input: Small errors in ‘a’, ‘b’, or especially ‘C’ can lead to different results for ‘c’.
- Angle Measurement: The calculator assumes angle C is in degrees. Make sure your input matches.
Frequently Asked Questions (FAQ)
- What if my triangle is right-angled?
- If angle C is 90 degrees, cos(90) = 0, and the Law of Cosines reduces to c² = a² + b², the Pythagorean theorem. Our find side of a triangle not right calculator handles this, but a Pythagorean Theorem calculator might be more direct.
- Can I find angles using this calculator?
- No, this specific find side of a triangle not right calculator is designed to find a side given two sides and the included angle (SAS). To find angles, you’d rearrange the Law of Cosines or use the Law of Sines, often needing different inputs (like SSS or ASA). You might need a Law of Sines calculator or a full triangle solver.
- What is the Law of Cosines?
- It’s 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 a generalization of the Pythagorean theorem.
- What if my angle is greater than 180 degrees or negative?
- Angles in a triangle are always between 0 and 180 degrees (exclusive of 0 and 180 for a non-degenerate triangle). The calculator will show an error for angles outside this range.
- What units should I use for the sides?
- You can use any unit (meters, feet, cm, etc.), as long as you are consistent for sides ‘a’ and ‘b’. Side ‘c’ will be in the same unit.
- Does this calculator work for any triangle?
- Yes, the Law of Cosines works for any triangle, whether acute, obtuse, or right-angled, as long as you have the SAS (Side-Angle-Side) information.
- What is the ‘SAS’ case?
- SAS stands for “Side-Angle-Side,” meaning you know the lengths of two sides and the measure of the angle between them. This is the information required by our find side of a triangle not right calculator.
- What about the Law of Sines?
- The Law of Sines (a/sin(A) = b/sin(B) = c/sin(C)) is another rule for non-right triangles, useful when you have different information, like two angles and one side (AAS or ASA) or two sides and a non-included angle (SSA – which can be ambiguous). Our Law of Sines calculator can help there.
Related Tools and Internal Resources
- Right Triangle Calculator: If your triangle has a 90-degree angle.
- Law of Sines Calculator: For solving triangles with ASA, AAS, or SSA data.
- Triangle Area Calculator: Calculate the area using various formulas.
- Pythagorean Theorem Calculator: Specifically for right-angled triangles.
- Geometry Calculators: A collection of calculators for various geometric shapes.
- Math Solvers: Other mathematical tools and solvers.