Find Third Triangle Side Calculator

Find Third Triangle Side Calculator

Calculate the length of the third side of a triangle when you know two sides and the angle between them (SAS case) using the Law of Cosines. Our find third triangle side calculator is fast and accurate.

Formula Used (Law of Cosines): c = √(a² + b² – 2ab cos(C))

Summary of inputs and the calculated third side.

Input/Output	Value	Unit
Side a	10	units
Side b	15	units
Angle C	60	degrees
Calculated Side c	–	units

What is a Find Third Triangle Side Calculator?

A find third triangle side calculator is a tool used 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 (the SAS – Side-Angle-Side case). It primarily uses the Law of Cosines to perform this calculation. This calculator is particularly useful in geometry, trigonometry, engineering, and physics when you need to find an unknown side of a non-right-angled triangle (though it also works for right-angled triangles if the angle is 90 degrees).

Anyone studying or working with triangles, including students, teachers, engineers, architects, and surveyors, can benefit from using a find third triangle side calculator. It simplifies complex calculations and provides quick, accurate results.

A common misconception is that you can always use the Pythagorean theorem (a² + b² = c²) to find the third side. However, the Pythagorean theorem ONLY applies to right-angled triangles. For other triangles, the Law of Cosines is the general formula, and our find third triangle side calculator implements this.

Find Third Triangle Side Formula and Mathematical Explanation

To find the third side of a triangle when you know two sides (let's call them 'a' and 'b') and the angle between them (angle 'C'), we use the Law of Cosines. The formula is:

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 two known sides.
• 'C' is the known angle between sides 'a' and 'b', measured in degrees or radians (our calculator takes degrees and converts).
• '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 third triangle side calculator performs these steps:

1. Takes the lengths of side 'a' and side 'b'.
2. Takes the measure of angle 'C' in degrees.
3. Converts angle 'C' from degrees to radians because trigonometric functions in JavaScript (like `Math.cos()`) use radians (Radians = Degrees * π / 180).
4. Calculates a², b², and cos(C).
5. Calculates 2ab cos(C).
6. Calculates c² = a² + b² – 2ab cos(C).
7. Calculates c by taking the square root of c².

If angle C is 90 degrees, cos(90°) = 0, and the formula simplifies to c² = a² + b², which is the Pythagorean theorem.

Variables used in the Law of Cosines for the find third triangle side calculator.

Variable	Meaning	Unit	Typical Range
a, b	Lengths of the two known sides	units (e.g., cm, m, inches)	Positive numbers
C	Angle between sides a and b	Degrees	0 < C < 180
c	Length of the third side (opposite angle C)	units (same as a, b)	Positive number
cos(C)	Cosine of angle C	Dimensionless	-1 to 1

Practical Examples (Real-World Use Cases)

Let's see how the find third triangle side calculator works with practical examples.

Example 1: Surveying a Plot of Land

A surveyor measures two sides of a triangular plot of land as 120 meters and 150 meters. The angle between these two sides is measured as 75 degrees. They need to find the length of the third side.

• Side a = 120 m
• Side b = 150 m
• Angle C = 75 degrees

Using the find third triangle side calculator (or the formula c = √(120² + 150² – 2 * 120 * 150 * cos(75°))):

c ≈ √(14400 + 22500 – 36000 * 0.2588) ≈ √(36900 – 9316.8) ≈ √(27583.2) ≈ 166.08 meters.

The third side is approximately 166.08 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 degrees

Using the find third triangle side calculator (c = √(50² + 70² – 2 * 50 * 70 * cos(110°))):

c ≈ √(2500 + 4900 – 7000 * (-0.3420)) ≈ √(7400 + 2394) ≈ √(9794) ≈ 98.96 nautical miles.

The ship is approximately 98.96 nautical miles from its starting point.

How to Use This Find Third Triangle Side Calculator

1. Enter Side a: Input the length of the first known side into the "Side a" field.
2. Enter Side b: Input the length of the second known side into the "Side b" field.
3. Enter Angle C: Input the angle between sides a and b, in degrees, into the "Angle C" field.
4. Calculate: The calculator will automatically update the results as you type, or you can click the "Calculate Side c" button.
5. Read Results: The primary result, "Side c," will be displayed prominently. You'll also see intermediate calculations and the formula used.
6. Analyze Chart and Table: The chart visually compares the side lengths, and the table summarizes inputs and outputs.
7. Reset: Click "Reset" to clear the fields and start over with default values.
8. Copy: Click "Copy Results" to copy the calculated side 'c' and other details to your clipboard.

The results from the find third triangle side calculator directly give you the length of the unknown side. This is crucial for planning, construction, or further calculations based on the triangle's dimensions.

Key Factors That Affect Find Third Triangle Side Results

The calculation of the third side using our find third triangle side calculator depends directly on:

1. Length of Side a: The longer side 'a' is, the longer side 'c' is likely to be, especially if angle C is obtuse.
2. Length of Side b: Similar to side 'a', the length of side 'b' directly influences the length of side 'c'.
3. Magnitude of Angle C: This is a crucial factor.
   • If C is close to 0 degrees, c will be close to |a – b|.
   • If C is 90 degrees (right angle), c = √(a² + b²) (Pythagorean theorem).
   • If C is close to 180 degrees, c will be close to a + b.
4. Units of Measurement: Ensure that sides 'a' and 'b' are entered in the same units. The resulting side 'c' will be in those same units. The calculator itself is unit-agnostic.
5. Accuracy of Input Values: Small errors in the input lengths or angle can lead to different results for side 'c'. Precise measurements are key.
6. Angle Measurement (Degrees vs. Radians): Our calculator expects the angle in degrees and converts it internally. If you are doing manual calculations, ensure your calculator or software is set to the correct mode (degrees or radians) for the cosine function.

Understanding these factors helps in interpreting the results from the find third triangle side calculator accurately.

Frequently Asked Questions (FAQ)

1. What is the Law of Cosines?

The Law of Cosines is 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 used by our find third triangle side calculator.

2. Can this calculator be used for any triangle?

Yes, this find third triangle side calculator uses the Law of Cosines, which is applicable to any triangle, whether it's acute, obtuse, or right-angled, as long as you know two sides and the included angle (SAS).

3. What if I know three sides and want to find an angle?

This calculator is specifically for finding the third side given two sides and the included angle. To find an angle given three sides (SSS), you would rearrange the Law of Cosines: C = arccos((a² + b² – c²) / 2ab). You might need a triangle angle calculator for that.

4. What if I know two angles and one side (ASA or AAS)?

If you know two angles and one side, you would first find the third angle (since angles sum to 180 degrees) and then use the Law of Sines to find the other sides. Our find third triangle side calculator focuses on the SAS case. You might look for a Law of Sines calculator.

5. Does the calculator handle units?

The calculator assumes the units for sides 'a' and 'b' are the same, and the result for side 'c' will be in those same units. It doesn't perform unit conversions.

6. What happens if I enter an angle of 90 degrees?

If you enter 90 degrees for Angle C, cos(90°) = 0, and the formula becomes c² = a² + b², which is the Pythagorean theorem. The calculator will correctly find the hypotenuse of a right triangle.

7. Can I enter negative values for sides or angles?

No, side lengths must be positive, and the angle C must be between 0 and 180 degrees (exclusive). The calculator includes validation to prevent invalid inputs.

8. Why is the Law of Cosines better than the Pythagorean theorem for this?

The Pythagorean theorem only works for right-angled triangles. The Law of Cosines works for ALL triangles, making the find third triangle side calculator more versatile. The Pythagorean theorem is just a special case of the Law of Cosines when the angle is 90 degrees.