Triangle Side Length Calculator (Find Length)
Calculate Side ‘c’ (using Cosine Rule)
Enter two sides and the angle between them (SAS) to find the third side.
Triangle Visualization (Approximate)
What is a Triangle Side Length Calculator?
A triangle side length calculator is a tool used to determine the length of an unknown side of a triangle when other information, such as the lengths of other sides and the measures of angles, is known. This calculator specifically helps you find the length of a side using various geometric principles, most commonly the Law of Cosines (or Cosine Rule) when you know two sides and the included angle (SAS), or the Law of Sines and other trigonometric relationships in different scenarios. Our triangle side length calculator is designed for ease of use and accuracy.
Anyone needing to find missing dimensions of a triangle can use this tool. This includes students learning trigonometry, engineers, architects, surveyors, and even DIY enthusiasts working on projects. A common misconception is that you always need a right-angled triangle, but a triangle side length calculator can often handle non-right-angled (oblique) triangles too using the Sine and Cosine Rules.
Triangle Side Length Calculator: Formula and Mathematical Explanation
When you know two sides of a triangle and the angle between them (Side-Angle-Side or SAS configuration), you can find the length of the third side using the Law of Cosines (also known as the Cosine Rule).
If you 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)
- b² = a² + c² – 2ac cos(B)
- a² = b² + c² – 2bc cos(A)
Our triangle side length calculator primarily uses the first formula (c² = a² + b² – 2ab cos(C)) to find side ‘c’ when ‘a’, ‘b’, and angle ‘C’ are known.
Step-by-step to find side ‘c’:
- Square the length of side ‘a’ (a²).
- Square the length of side ‘b’ (b²).
- Multiply 2 by side ‘a’ and side ‘b’ (2ab).
- Find the cosine of angle ‘C’ (cos(C)). Make sure ‘C’ is in radians if using programming functions, or use the degree mode on a calculator. Our calculator handles the conversion.
- Multiply the results from step 3 and step 4 (2ab cos(C)).
- Subtract the result of step 5 from the sum of step 1 and step 2 (a² + b² – 2ab cos(C)). This gives c².
- Take the square root of the result from step 6 to get the length of side ‘c’.
| Variable | Meaning | Unit | Typical Range |
|---|---|---|---|
| a, b | Lengths of the two known sides | Units of length (e.g., cm, m, inches) | Positive numbers |
| C | The angle included between sides ‘a’ and ‘b’ | Degrees or Radians | 0° < C < 180° (or 0 < C < π radians) |
| c | The length of the side opposite angle ‘C’ (the unknown side) | Same units as ‘a’ and ‘b’ | Positive number |
| cos(C) | The 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, with the angle between these two sides being 75 degrees. They need to find the length of the third side.
- Side a = 120 m
- Side b = 150 m
- Angle C = 75°
Using the triangle side length calculator (or the formula c² = a² + b² – 2ab cos(C)):
c² = 120² + 150² – 2 * 120 * 150 * cos(75°)
c² = 14400 + 22500 – 36000 * 0.2588
c² = 36900 – 9316.8 = 27583.2
c = √27583.2 ≈ 166.08 meters
The third side is approximately 166.08 meters long.
Example 2: Navigation
A ship sails 50 nautical miles on one bearing, then changes course and sails 70 nautical miles. If the angle between the two legs of the journey was 110 degrees, how far is the ship from its starting point?
- Side a = 50 nm
- Side b = 70 nm
- Angle C = 110°
Using our triangle side length calculator:
c² = 50² + 70² – 2 * 50 * 70 * cos(110°)
c² = 2500 + 4900 – 7000 * (-0.3420)
c² = 7400 + 2394 = 9794
c = √9794 ≈ 98.96 nautical miles
The ship is approximately 98.96 nautical miles from its start.
How to Use This Triangle Side Length 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 automatically updates as you type, or you can click “Calculate Side ‘c'”.
- Read Results: The primary result is the length of side ‘c’, displayed prominently. Intermediate calculations are also shown for clarity.
- Reset: Click “Reset” to clear the fields and start over with default values.
- Copy: Click “Copy Results” to copy the calculated lengths and angle to your clipboard.
The triangle side length calculator helps you quickly find the missing side, making complex calculations simple.
Key Factors That Affect Triangle Side Length Results
- Accuracy of Side Measurements: Small errors in measuring sides ‘a’ or ‘b’ will directly impact the calculated length of ‘c’. Ensure your initial measurements are as precise as possible.
- Accuracy of Angle Measurement: An error in measuring angle ‘C’ can significantly affect the result, especially because it’s used within a cosine function.
- Units Used: Ensure that both side ‘a’ and side ‘b’ are measured in the same units. The resulting side ‘c’ will be in those same units.
- Angle Being Between the Sides: The Cosine Rule (SAS) requires the angle ‘C’ to be the angle *between* the known sides ‘a’ and ‘b’. If you have a different angle, you might need the Sine Rule or other methods. Check out our sine rule calculator for other cases.
- 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. While this calculator finds ‘c’, it assumes ‘a’, ‘b’, and ‘C’ can form a valid triangle.
- Rounding: The number of decimal places used in intermediate calculations and the final result can slightly affect the precision. Our triangle side length calculator aims for reasonable precision.
Frequently Asked Questions (FAQ)
- What if I have two angles and one side (ASA or AAS)?
- You would typically use the Law of Sines first to find another side, then proceed. This specific triangle side length calculator focuses on the SAS case (Cosine Rule). You might need a more general triangle calculator.
- Can I use this calculator for a right-angled triangle?
- Yes, you can. If angle ‘C’ is 90 degrees, cos(90°) = 0, and the formula c² = a² + b² – 2ab * 0 simplifies to c² = a² + b², which is the Pythagorean theorem. However, a dedicated Pythagorean theorem calculator might be quicker for right triangles if you know two sides.
- What units should I use for sides and angles?
- Sides ‘a’ and ‘b’ should be in the same unit (e.g., cm, meters, inches). Angle ‘C’ must be in degrees for this calculator. The result ‘c’ will be in the same unit as ‘a’ and ‘b’.
- Why is the angle limited to 0-179.99 degrees?
- The sum of angles in any triangle is 180 degrees. A single angle in a triangle must be greater than 0 and less than 180 degrees to form a valid triangle.
- 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. It’s essential for solving triangles that are not right-angled when you have SAS or SSS information. Our triangle side length calculator is based on it.
- What if I know all three sides (SSS) and want to find an angle?
- You can rearrange the Law of Cosines to solve for the angle. For example, cos(C) = (a² + b² – c²) / 2ab. You’d then use arccos to find C.
- Does this calculator find the area?
- No, this calculator focuses on finding a side length. For area, especially with SAS, you can use the formula Area = 0.5 * a * b * sin(C). See our triangle area calculator.
- How accurate is this triangle side length calculator?
- The calculator uses standard mathematical formulas and is accurate based on the precision of your input values and the internal calculations (typically to many decimal places before final rounding).
Related Tools and Internal Resources
- General Triangle Calculator: Solves various triangle problems given different inputs.
- Pythagorean Theorem Calculator: Specifically for right-angled triangles to find sides.
- Sine and Cosine Rule Calculator: Explore the Law of Sines and Cosines for non-right triangles.
- Triangle Area Calculator: Calculate the area of a triangle using different formulas.
- Geometry Calculators: A collection of tools for various geometric shapes.
- Math Calculators: Our main hub for mathematical and financial calculators.