Find Side C Of A Triangle Calculator

Easily calculate the length of side ‘c’ of a triangle given the lengths of sides ‘a’ and ‘b’ and the angle ‘C’ between them using our find side c of a triangle calculator based on the Law of Cosines.

Calculator

Understanding the Results

Angle C (Degrees)	Angle C (Radians)	cos(C)	Side c Length

Table showing side c for various angles C with sides a and b fixed.

What is a Find Side c of a Triangle Calculator?

A “find side c of a triangle calculator” is a tool used to determine the length of the third side (c) of a triangle when you know the lengths of the other two sides (a and b) and the angle (C) between them. This scenario is often referred to as SAS (Side-Angle-Side). The calculator employs 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.

This calculator is particularly useful for students of geometry and trigonometry, engineers, architects, surveyors, and anyone needing to solve for an unknown side of a non-right-angled triangle given two sides and the included angle. It bypasses the need for manual calculations, providing quick and accurate results.

Common misconceptions include thinking it only works for right-angled triangles (that’s where the Pythagorean theorem is more direct, though the Law of Cosines still applies) or that any three known values of a triangle can be used (this specific calculator requires two sides and the *included* angle).

Find Side c of a Triangle Calculator Formula and Mathematical Explanation

The core of the find side c of a triangle calculator is the Law of Cosines. The formula is:

c² = a² + b² – 2ab * cos(C)

From this, we find side c by taking the square root:

c = √(a² + b² – 2ab * cos(C))

Here’s a step-by-step breakdown:

1. Square the lengths of sides a and b: Calculate a² and b².
2. Convert Angle C to Radians: The angle C provided in degrees must be converted to radians because trigonometric functions in most programming languages (and mathematically) use radians. Radians = Degrees * (π / 180).
3. Calculate the Cosine of Angle C: Find cos(C) using the angle in radians.
4. Multiply 2ab * cos(C): Calculate the product of 2, side a, side b, and cos(C).
5. Complete the Equation for c²: Substitute the values into c² = a² + b² – 2ab * cos(C).
6. Find c: Take the square root of c² to get the length of side c.

Variables Used in the Law of Cosines

Variable	Meaning	Unit	Typical Range
a	Length of side a	Length units (e.g., m, cm, ft)	Positive numbers
b	Length of side b	Length units (e.g., m, cm, ft)	Positive numbers
C	Angle included between sides a and b	Degrees or Radians	0° < C < 180° (or 0 < C < π radians)
c	Length of side c (opposite angle C)	Length units (e.g., m, cm, ft)	Positive numbers
cos(C)	Cosine of angle C	Dimensionless	-1 to 1

Practical Examples (Real-World Use Cases)

The find side c of a triangle calculator, based on the Law of Cosines, is invaluable in various real-world scenarios:

Example 1: Land Surveying

A surveyor needs to determine the distance across a lake between two points, A and B. They set up a point C on land and measure the distance from C to A as 150 meters (side b) and from C to B as 200 meters (side a). The angle ACB is measured to be 55 degrees (angle C). Using the calculator:

• Side a = 200 m
• Side b = 150 m
• Angle C = 55°

Inputting these values, the calculator would find side c (the distance AB across the lake) to be approximately 170.8 meters.

Example 2: Navigation

A pilot flies 100 miles due east, then turns 110 degrees to the north of east and flies another 80 miles. How far is the plane from its starting point? Here, the first leg is side a (100 miles), the second leg is side b (80 miles), and the angle *inside* the triangle formed is 180° – 110° = 70° (angle C).

• Side a = 100 miles
• Side b = 80 miles
• Angle C = 70°

The find side c of a triangle calculator would determine side c, the direct distance from the start, is approximately 104.1 miles.

How to Use This Find Side c of a Triangle Calculator

Using our find side c of a triangle calculator is straightforward:

1. Enter Side a: Input the length of one of the known sides into the “Length of Side a” field. Ensure it’s a positive number.
2. Enter Side b: Input the length of the other known side into the “Length of Side b” field. This also must be a positive number.
3. Enter Angle C: Input the angle between sides a and b, in degrees, into the “Angle C” field. The angle must be between 0 and 180 degrees (exclusive of 0 and 180 for a valid triangle with the Law of Cosines used this way, though our calculator limits to 1-179).
4. Calculate: Click the “Calculate Side c” button, or the results will update automatically if you change input values after the first calculation.
5. Read the Results: The calculator will display:
   • The length of side c (primary result).
   • Intermediate values like angle C in radians, cos(C), and c².
6. Interpret the Chart and Table: The dynamic chart and table show how side c varies with angle C, keeping sides a and b constant at your entered values, giving you a visual understanding.
7. Reset: Click “Reset” to clear the fields to their default values.
8. Copy: Click “Copy Results” to copy the main result and intermediate values to your clipboard.

Decision-making: If you are designing something or surveying, the calculated side c gives you the precise length needed. If the result seems unexpected, double-check your input values, especially the angle measurement.

Key Factors That Affect Find Side c of a Triangle Calculator Results

The length of side c is determined by three key factors:

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: Similar to side a, the length of side b directly influences the length of side c.
3. Magnitude of Angle C: This is crucial.
   • As angle C approaches 0°, cos(C) approaches 1, and c² approaches (a-b)², so c approaches |a-b|.
   • As angle C approaches 90° (a right angle), cos(C) becomes 0, and the formula reduces to c² = a² + b² (the Pythagorean theorem).
   • As angle C approaches 180°, cos(C) approaches -1, and c² approaches (a+b)², so c approaches a+b.
4. Units of Measurement: The unit of side c will be the same as the units used for sides a and b. Ensure consistency.
5. Accuracy of Input: Small errors in measuring sides a, b, or especially angle C can lead to larger errors in the calculated side c, particularly when the angle is very small or very close to 180 degrees.
6. 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 given a, b, and C, the results imply c < a+b.

Frequently Asked Questions (FAQ)

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). Our find side c of a triangle calculator uses this law.

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

Yes. If angle C is 90 degrees, cos(90°) = 0, and the formula simplifies to c² = a² + b², which is the Pythagorean theorem. So, the find side c of a triangle calculator works perfectly for right triangles too if you know the two sides forming the right angle.

What units should I use for the sides?

You can use any unit of length (meters, feet, inches, cm, etc.) for sides a and b, as long as you use the SAME unit for both. The result for side c will be in that same unit.

What if my angle is 0 or 180 degrees?

A triangle cannot have an angle of 0 or 180 degrees. If the angle were 0, the sides would lie on top of each other, and c would be |a-b|. If it were 180, they would form a straight line, and c would be a+b. Our calculator restricts the angle to 1-179 degrees for a valid triangle.

Does this find side c of a triangle calculator work if I know other angles or sides?

This specific calculator is for the SAS (Side-Angle-Side) case – when you know two sides and the angle *between* them. If you know other combinations (like ASA, SSS, AAS), you might need the Law of Sines or other triangle-solving methods. See our triangle solver for more options.

Why is angle C converted to radians?

Most mathematical and programming functions for cosine (cos) expect the angle to be in radians, not degrees. The calculator converts the degree input to radians before calculating the cosine.

How accurate is this find side c of a triangle calculator?

The calculator uses standard mathematical formulas and is as accurate as the input values you provide. Ensure your measurements for sides a, b, and angle C are precise for an accurate result for side c.

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

You can rearrange the Law of Cosines to solve for an angle if you know all three sides: cos(C) = (a² + b² – c²) / 2ab. You would then use the inverse cosine (arccos) function. Or use our angle calculator.

Related Tools and Internal Resources

Explore other calculators and resources related to triangles and geometry:

• Law of Sines Calculator: Use when you know AAS (Angle-Angle-Side) or SSA (Side-Side-Angle – the ambiguous case).
• Triangle Area Calculator: Calculate the area of a triangle using various formulas (including base-height, Heron’s formula, or using trigonometry).
• Pythagorean Theorem Calculator: Specifically for right-angled triangles to find a side given the other two.
• Angle Calculator: Find angles in a triangle given sides, or convert between angle units.
• Triangle Solver: A comprehensive tool to solve triangles given various known sides and angles.
• Geometry Formulas: A collection of important geometry formulas, including those for triangles.