Find the Side of a Triangle Calculator
Easily calculate the missing side of a triangle using our Find the Side of a Triangle Calculator based on the Law of Cosines and Sines. Input two sides and an angle, or other combinations, to find the unknown side.
Triangle Side Calculator
Angle 'A': – degrees
Angle 'B': – degrees
Area: – square units
Using the Law of Cosines: c² = a² + b² – 2ab cos(C), then Law of Sines for angles A and B, and Area = 0.5 * a * b * sin(C).
Side 'c' Variation with Angle 'C'
| Angle C (degrees) | Side c |
|---|---|
| Results will appear here. | |
Side 'c' vs Angle 'C' Chart
What is a Find the Side of a Triangle Calculator?
A Find the Side of a Triangle Calculator is a tool used to determine the length of an unknown side of a triangle when you know the lengths of other sides and/or the measure of some angles. It commonly employs the Law of Cosines or the Law of Sines, fundamental theorems in trigonometry, to perform these calculations. Our Find the Side of a Triangle Calculator is particularly useful when you know two sides and the included angle (SAS), or other combinations.
Anyone studying geometry, trigonometry, or involved in fields like engineering, architecture, physics, or even surveying can benefit from using a Find the Side of a Triangle Calculator. It automates complex calculations, saving time and reducing the risk of manual errors. For example, if you know two sides of a triangular piece of land and the angle between them, you can find the length of the third side using this calculator.
A common misconception is that you always need three pieces of information to find a side, but it depends on what you know. For instance, knowing two angles and one side (AAS or ASA) or two sides and one angle (SAS or SSA – the ambiguous case) allows you to find other sides and angles, often with the help of a Find the Side of a Triangle Calculator.
Find the Side of a Triangle Calculator Formula and Mathematical Explanation
The primary formula used by our Find the Side of a Triangle Calculator when two sides and the included angle are known is the Law of Cosines:
c² = a² + b² – 2ab cos(C)
Where 'a' and 'b' are the lengths of two sides, 'C' is the angle between them (in radians for the formula), and 'c' is the length of the side opposite angle C.
Once side 'c' is found, the Law of Sines can be used to find the other angles:
a/sin(A) = b/sin(B) = c/sin(C)
From which we can derive:
sin(A) = (a * sin(C)) / c
sin(B) = (b * sin(C)) / c
A = asin( (a * sin(C)) / c )
B = asin( (b * sin(C)) / c )
The area of the triangle can be calculated as:
Area = 0.5 * a * b * sin(C)
Here's a breakdown of the variables:
| Variable | Meaning | Unit | Typical Range |
|---|---|---|---|
| a, b, c | Lengths of the sides of the triangle | Units of length (e.g., cm, m, inches) | > 0 |
| A, B, C | Angles opposite sides a, b, c respectively | Degrees (or radians in formulas) | 0° < Angle < 180° |
| Area | Area of the triangle | Square units | > 0 |
Our Find the Side of a Triangle Calculator handles the conversion between degrees and radians automatically.
Practical Examples (Real-World Use Cases)
Example 1: Surveying
A surveyor measures two sides of a triangular plot of land as 100 meters and 150 meters, and the angle between these two sides is 70 degrees. They need to find the length of the third side.
- Side a = 100 m
- Side b = 150 m
- Angle C = 70°
Using the Find the Side of a Triangle Calculator (or the Law of Cosines):
c² = 100² + 150² – 2 * 100 * 150 * cos(70°)
c² = 10000 + 22500 – 30000 * 0.34202
c² = 32500 – 10260.6
c² = 22239.4
c ≈ 149.13 meters
The third side is approximately 149.13 meters.
Example 2: Navigation
A ship sails 50 nautical miles on a bearing, then changes course and sails 70 nautical miles. 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 the Find the Side of a Triangle Calculator:
c² = 50² + 70² – 2 * 50 * 70 * cos(110°)
c² = 2500 + 4900 – 7000 * (-0.34202)
c² = 7400 + 2394.14
c² = 9794.14
c ≈ 98.96 nautical miles
The ship is approximately 98.96 nautical miles from its start.
How to Use This Find the Side of a Triangle Calculator
Using our Find the Side of a Triangle Calculator is straightforward:
- Enter Side 'a': Input the length of one known side.
- Enter Side 'b': Input the length of the second known side.
- Enter Angle 'C': Input the measure of the angle between sides 'a' and 'b' in degrees.
- Calculate: Click the "Calculate Side 'c'" button or simply change the input values. The calculator updates automatically.
- Read Results: The primary result is the length of Side 'c'. You will also see the calculated values for Angle 'A', Angle 'B', and the Area of the triangle.
- Check Table and Chart: The table and chart below the calculator show how Side 'c' varies with Angle 'C' for the given 'a' and 'b', providing more insight. Our Triangle Area Calculator can give more area details.
The results give you the length of the unknown side and other triangle properties. You can use the "Reset" button to clear inputs and "Copy Results" to copy the main findings.
Key Factors That Affect Triangle Side Lengths
The length of a side of a triangle is determined by the lengths of the other sides and the angles of the triangle. Here are key factors:
- Lengths of Other Sides: The longer the other two sides are, the longer the third side can potentially be, depending on the included angle.
- Included Angle: The angle between two known sides significantly impacts the length of the third side. A larger angle (up to 180°) generally results in a longer third side (as seen in the Law of Cosines).
- Other Angles: If you know angles and one side, the Law of Sines shows how the side lengths are proportional to the sines of the opposite angles.
- Triangle Inequality Theorem: The sum of the lengths of any two sides of a triangle must be greater than the length of the third side. This constrains the possible lengths.
- Type of Triangle: Whether it's a right-angled, isosceles, or equilateral triangle imposes specific relationships between sides and angles. Our Right Triangle Calculator is specialized for that case.
- Measurement Units: Ensure all side lengths are in the same units for the calculation to be correct. The units of the calculated side will be the same as the input sides.
Understanding these factors helps in using any Find the Side of a Triangle Calculator effectively and interpreting the results in context. For more on the laws used, see our pages on the Law of Sines and Law of Cosines.
Frequently Asked Questions (FAQ)
- Q1: What information do I need to use the Find the Side of a Triangle Calculator?
- A1: For this specific version, you need the lengths of two sides and the measure of the angle between them (SAS – Side-Angle-Side) to find the third side using the Law of Cosines. Other versions might use the Law of Sines with different inputs (like ASA or AAS).
- Q2: Can I find a side if I know all three angles?
- A2: No, knowing only the three angles determines the shape of the triangle but not its size. You need at least one side length along with angles to find other sides.
- Q3: What is the Law of Cosines?
- A3: The Law of Cosines relates the lengths of the sides of a triangle to the cosine of one of its angles: c² = a² + b² – 2ab cos(C). Our Find the Side of a Triangle Calculator uses this.
- Q4: What is the Law of Sines?
- A4: The Law of Sines relates the lengths of the sides of a triangle to the sines of its angles: a/sin(A) = b/sin(B) = c/sin(C). It's useful after finding one side with the Law of Cosines, or if you have ASA or AAS information.
- Q5: What if the angle I enter is 0 or 180 degrees?
- A5: The calculator expects an angle between 0 and 180 degrees (exclusive) because a triangle cannot have an angle of 0 or 180 degrees. The calculator will show an error for such inputs.
- Q6: What if my input sides are negative or zero?
- A6: Side lengths must be positive. The calculator will prompt you to enter positive values if you input zero or negative numbers.
- Q7: Can this calculator handle the 'ambiguous case' (SSA)?
- A7: This particular calculator is set up for SAS (two sides and the included angle). The SSA case (two sides and a non-included angle) can result in zero, one, or two possible triangles and requires a different approach or calculator, like a triangle angle calculator used in reverse, or a dedicated SSA solver.
- Q8: How accurate is the Find the Side of a Triangle Calculator?
- A8: The calculator uses standard trigonometric formulas and is very accurate, provided the input values are precise. The results are typically rounded to a few decimal places.