Find Third Side Of Triangle Calculator

Calculate the Third Side

Enter the lengths of two sides and the angle between them (in degrees) to find the third side using the Law of Cosines.

Input	Value
Side ‘a’
Side ‘b’
Angle ‘C’ (degrees)
Side ‘c’

Summary of inputs and calculated third side.

What is a Find Third Side of Triangle Calculator?

A find third side of triangle calculator is a tool designed to calculate the length of the unknown third side of a triangle when you know the lengths of the other two sides and the angle between them. It primarily uses the Law of Cosines to perform this calculation. This is particularly useful in geometry, trigonometry, engineering, and various fields where triangle properties are important.

Anyone needing to solve for triangle dimensions, such as students, engineers, architects, or surveyors, can use a find third side of triangle calculator. A common misconception is that you always need a right-angled triangle to find sides; however, this calculator (using the Law of Cosines) works for any triangle, not just right-angled ones.

Find Third Side of Triangle Formula (Law of Cosines) and Mathematical Explanation

To find the third side of a triangle when you know two sides (a and b) and the included angle (C), we use the Law of Cosines:

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’.
• 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))

Step-by-step Derivation:

1. Square the length of side ‘a’ (a²).
2. Square the length of side ‘b’ (b²).
3. Calculate the cosine of angle ‘C’ (cos(C)). Make sure ‘C’ is in radians if your calculator function requires it, or convert from degrees. Our calculator handles degree input.
4. Multiply 2, ‘a’, ‘b’, and cos(C) together (2ab cos(C)).
5. Subtract the result from step 4 from the sum of the results from steps 1 and 2 (a² + b² – 2ab cos(C)). This gives c².
6. Take the square root of the result from step 5 to find ‘c’.

Variables Table:

Variable	Meaning	Unit	Typical Range
a	Length of the first known side	Units (e.g., cm, m, inches)	> 0
b	Length of the second known side	Units (e.g., cm, m, inches)	> 0
C	Angle between sides ‘a’ and ‘b’	Degrees	0 < C < 180
c	Length of the third side (opposite angle C)	Units (e.g., cm, m, inches)	> 0

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

Practical Examples (Real-World Use Cases)

Example 1: Surveying

A surveyor measures two sides of a triangular plot of land as 120 meters and 150 meters, with the angle between them being 75 degrees. They need to find the length of the third side.

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

Using the find third side of triangle calculator (or formula):
c² = 120² + 150² – 2 * 120 * 150 * cos(75°)
c² = 14400 + 22500 – 36000 * 0.2588
c² = 36900 – 9316.8
c² = 27583.2
c = √27583.2 ≈ 166.08 meters

The third side is approximately 166.08 meters long.

Example 2: Navigation

A ship sails 30 miles East, then turns 110 degrees North of East and sails 40 miles. How far is the ship from its starting point?

• a = 30 miles
• b = 40 miles
• C = 110 degrees

Using the find third side of triangle calculator:
c² = 30² + 40² – 2 * 30 * 40 * cos(110°)
c² = 900 + 1600 – 2400 * (-0.3420)
c² = 2500 + 820.8
c² = 3320.8
c = √3320.8 ≈ 57.63 miles

The ship is approximately 57.63 miles from its starting point.

How to Use This Find Third Side of Triangle Calculator

1. Enter Side ‘a’ Length: Input the length of one of the known sides into the “Side ‘a’ Length” field.
2. Enter Side ‘b’ Length: Input the length of the other known side into the “Side ‘b’ Length” field.
3. Enter Angle ‘C’: Input the angle (in degrees) between sides ‘a’ and ‘b’ into the “Angle ‘C'” field. Ensure this angle is between 0 and 180 degrees.
4. Calculate: Click the “Calculate” button (or the results will update automatically if you change values). The calculator will display the length of the third side ‘c’, along with intermediate steps.
5. Read Results: The primary result is the length of side ‘c’. Intermediate values like a², b², 2ab cos(C), and c² are also shown for clarity. A table and a chart will also visualize the inputs and output.
6. Reset: Click “Reset” to clear the fields and start over with default values.

This find third side of triangle calculator simplifies the application of the Law of Cosines.

Key Factors That Affect the Third Side Calculation

1. Length of Side ‘a’: The longer side ‘a’ is, the longer side ‘c’ is likely to be, especially if angle C is large.
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 crucial.
   • If C = 90 degrees (right angle), cos(C) = 0, and the formula becomes c² = a² + b² (Pythagorean theorem).
   • If C < 90 degrees (acute), cos(C) > 0, reducing the value of c² compared to a² + b².
   • If C > 90 degrees (obtuse), cos(C) < 0, increasing the value of c² compared to a² + b².
4. Units of Measurement: Ensure both side ‘a’ and ‘b’ are in the same units. The resulting side ‘c’ will also be in those units.
5. Accuracy of Input: Small errors in the input values, especially the angle, can lead to larger errors in the calculated side ‘c’, particularly when sides are long or the angle is close to 0 or 180 degrees.
6. Rounding: The precision of the cosine value and the final square root can affect the result’s accuracy. Our find third side of triangle calculator aims for high precision.

Frequently Asked Questions (FAQ)

1. What formula does the find third side of triangle calculator use?

It primarily uses the Law of Cosines: c² = a² + b² – 2ab cos(C).

2. Can I use this calculator for a right-angled triangle?

Yes. If you enter 90 degrees for angle C, the term -2ab cos(90°) becomes zero, and it simplifies to the Pythagorean theorem c² = a² + b².

3. What if I know two angles and one side, or two sides and a non-included angle?

For those cases, you would typically use the Law of Sines first, possibly in conjunction with the fact that angles in a triangle sum to 180 degrees. This calculator is specifically for the two sides and included angle case using the Law of Cosines.

4. Do I need to enter the angle in radians?

No, our find third side of triangle calculator accepts the angle in degrees and converts it internally for the cosine calculation.

5. What happens if I enter an angle greater than 180 degrees or less than 0?

The calculator will show an error message as a valid angle within a triangle must be between 0 and 180 degrees (exclusive of 0 and 180 in most practical triangle cases when given two sides).

6. Why is it called the Law of Cosines?

Because it relates the lengths of the sides of a triangle to the cosine of one of its angles.

7. Can the calculated third side be negative?

No, the length of a side of a triangle must always be a positive value. The formula involves squares and a square root, ensuring a positive result for the length ‘c’.

8. What are the units for the third side?

The units for the third side ‘c’ will be the same as the units you used for sides ‘a’ and ‘b’. If you used meters, the result is in meters.

Related Tools and Internal Resources

• Law of Sines Calculator: Use this if you know two angles and a side, or two sides and a non-included angle.
• Triangle Area Calculator: Calculate the area of a triangle using various formulas.
• Pythagorean Theorem Calculator: Specifically for right-angled triangles to find a missing side.
• Angle Calculator: Tools for angle conversions and calculations.
• Geometry Formulas: A collection of useful formulas in geometry.
• Math Calculators: Our main hub for various mathematical calculators.

Our find third side of triangle calculator is one of many tools we offer to help with geometry and trigonometry problems.