Find Side C Calculator

Easily calculate the length of side ‘c’ of a triangle given the lengths of sides ‘a’, ‘b’, and the angle ‘C’ between them using our Find Side c Calculator.

Calculator

Example Values of Side ‘c’

Angle C (degrees)	Side c (for a=3, b=4)
0	1.00
30	2.05
60	3.61
90	5.00
120	6.08
150	6.77
180	7.00

Table showing calculated side ‘c’ for fixed sides a=3, b=4 and varying Angle C.

What is a Find Side c Calculator?

A Find Side c Calculator is a tool used to determine the length of the third side (‘c’) of any triangle when the lengths of the other two sides (‘a’ and ‘b’) and the angle (‘C’) between them are known. It primarily uses the Law of Cosines, a fundamental theorem in trigonometry that relates the lengths of the sides of a triangle to the cosine of one of its angles. Our Find Side c Calculator is especially useful for non-right-angled triangles, where the Pythagorean theorem (a² + b² = c²) doesn’t directly apply (though the Law of Cosines becomes the Pythagorean theorem when angle C is 90 degrees).

Anyone studying geometry, trigonometry, or working in fields like engineering, surveying, navigation, or physics might use a Find Side c Calculator. It’s helpful whenever you have two sides and the included angle and need to find the opposite side. Common misconceptions are that it only works for right triangles (it works for all triangles) or that it’s the same as the Pythagorean theorem (it’s a generalization of it).

Find Side c Calculator Formula and Mathematical Explanation

The Find Side c Calculator uses 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 other two sides.
• ‘C’ is the angle between sides ‘a’ and ‘b’.
• cos(C) is the cosine of angle ‘C’. Note that the angle C must be in radians when using most programming language math functions, but our calculator takes it in degrees and converts it.

To find ‘c’, we take the square root of the result:

c = √(a² + b² – 2ab * cos(C))

The derivation involves dropping a perpendicular from one vertex to the opposite side and using the Pythagorean theorem and trigonometric definitions.

Variables Table

Variable	Meaning	Unit	Typical Range
a	Length of side a	Length units (e.g., cm, m, inches)	> 0
b	Length of side b	Length units (e.g., cm, m, inches)	> 0
C	Angle between sides a and b	Degrees	0 < C < 180
c	Length of side c (opposite angle C)	Length units (e.g., cm, m, inches)	> 0

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 included angle between them being 75 degrees. To find the length of the third side (‘c’) of the plot, they use the Find Side c Calculator (Law of Cosines).

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

c = √(120² + 150² – 2 * 120 * 150 * cos(75°))
c = √(14400 + 22500 – 36000 * 0.2588)
c = √(36900 – 9316.8)
c = √27583.2 ≈ 166.08 meters

The third side is approximately 166.08 meters long.

Example 2: Navigation

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

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

c = √(30² + 40² – 2 * 30 * 40 * cos(110°))
c = √(900 + 1600 – 2400 * (-0.3420))
c = √(2500 + 820.8)
c = √3320.8 ≈ 57.63 nautical miles

The ship is approximately 57.63 nautical miles from its starting point.

How to Use This Find Side c Calculator

1. Enter Side ‘a’: Input the length of one of the sides adjacent to angle C into the “Length of Side ‘a'” field.
2. Enter Side ‘b’: Input the length of the other side adjacent to angle C into the “Length of Side ‘b'” field.
3. Enter Angle ‘C’: Input the angle between sides ‘a’ and ‘b’ in degrees into the “Angle ‘C'” field.
4. View Results: The calculator automatically updates the length of side ‘c’, along with intermediate values like a², b², cos(C), and 2ab*cos(C). The primary result for side ‘c’ is highlighted.
5. Use the Chart: The chart below the calculator visualizes how side ‘c’ changes as you vary Angle ‘C’ (from 0 to 180 degrees), keeping the entered ‘a’ and ‘b’ values constant.
6. Reset: Click the “Reset” button to return to default values (a=3, b=4, C=90).
7. Copy Results: Click “Copy Results” to copy the main result, intermediates, and formula to your clipboard.

The Find Side c Calculator is a direct application of the Law of Cosines. It’s very useful when you have SAS (Side-Angle-Side) information for a triangle.

Key Factors That Affect Side c Results

The length of side ‘c’ calculated by the Find Side c Calculator is directly influenced by:

1. Length of Side ‘a’: The longer side ‘a’ is, the longer side ‘c’ will generally be, especially if angle C is large.
2. Length of Side ‘b’: Similarly, a longer side ‘b’ leads to a longer side ‘c’, particularly with larger angles C.
3. Magnitude of Angle ‘C’: This is crucial.
   • If C is small (near 0 degrees), ‘c’ will be close to |a – b|.
   • If C is 90 degrees (right angle), cos(C) = 0, and c² = a² + b² (Pythagorean theorem), so c = √(a² + b²).
   • If C is large (near 180 degrees), cos(C) is near -1, and c² is close to a² + b² + 2ab = (a+b)², so ‘c’ is close to a + b.
4. Units Used: The unit of side ‘c’ will be the same as the units used for sides ‘a’ and ‘b’. Consistency is key.
5. Accuracy of Input: Small errors in ‘a’, ‘b’, or especially ‘C’ can lead to different results for ‘c’.
6. Triangle Inequality: The sum of any two sides of a triangle must be greater than the third side. The calculator assumes valid triangle inputs, but the relationship between a, b, and C determines if a triangle can even be formed with the resulting ‘c’.

Understanding these factors helps in interpreting the results from the Find Side c Calculator and applying them correctly.

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 any triangle to the cosine of one of its angles: c² = a² + b² – 2ab cos(C). Our Find Side c Calculator is based on this law.

2. Can I use this Find Side c Calculator for right-angled triangles?

Yes. If you input an angle C of 90 degrees, cos(90°) = 0, and the formula simplifies to c² = a² + b², which is the Pythagorean theorem for right-angled triangles where ‘c’ is the hypotenuse.

3. What units should I use for sides ‘a’ and ‘b’?

You can use any consistent units of length (e.g., meters, feet, inches, centimeters). The resulting side ‘c’ will be in the same units.

4. What is the range for Angle C?

In a triangle, an angle must be greater than 0 and less than 180 degrees. The calculator is designed for this range.

5. What if I have two angles and one side?

If you have two angles and one side, you would first find the third angle (sum of angles = 180°) and then use the Law of Sines Calculator to find the other sides.

6. Is the Find Side c Calculator the same as a Triangle Solver?

It’s part of a Triangle Solver. A full triangle solver can find all sides and angles given sufficient information (like SAS, SSS, ASA, AAS).

7. Why is it called ‘side c’ and ‘angle C’?

By convention, in triangle notation, the side opposite a vertex (and its angle) is labeled with the corresponding lowercase letter. So, side ‘c’ is opposite angle ‘C’.

8. Can the result for side ‘c’ be negative?

No, the length of a side of a triangle must always be positive. The formula involves a square root of a value that should be positive for valid triangles and angles between 0 and 180 degrees.