Find Missing Side Of Triangle Calculator 3 Sides

Use this calculator to find the missing side ‘c’ of a triangle when you know sides ‘a’, ‘b’, and the angle ‘C’ between them (SAS – Side-Angle-Side).

Angle C (degrees)	Side c

Table showing calculated side ‘c’ for different values of Angle C.

What is a Find Missing Side of Triangle Calculator?

A find missing side of triangle calculator is a tool used to determine the length of an unknown side of a triangle when sufficient information about the other sides and/or angles is provided. Specifically, when you know two sides and the included angle (the angle between those two sides), this calculator often employs the Law of Cosines to find the third side. It’s invaluable for students, engineers, architects, and anyone working with geometric problems involving triangles.

This particular calculator focuses on the scenario where you have two sides (let’s call them ‘a’ and ‘b’) and the angle ‘C’ between them, and you want to find the length of the side ‘c’ opposite angle C. This is a classic application of the Law of Cosines, a fundamental theorem in trigonometry.

Who Should Use It?

• Students: Learning trigonometry and geometry can use it to verify their manual calculations.
• Engineers & Architects: For design and structural analysis involving triangular components.
• Surveyors: To calculate distances and boundaries.
• Game Developers: For physics and collision detection in virtual environments.

Common Misconceptions

One common misconception is that you can always find a missing side with any three pieces of information. However, you need specific combinations, like two sides and the included angle (SAS), three sides (SSS – to find angles), or two angles and a side (ASA or AAS), to uniquely define a triangle and find missing parts using standard laws like the Law of Sines or Cosines. A find missing side of triangle calculator is most straightforward when using the Law of Cosines for the SAS case.

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

When you know two sides of a triangle (a and b) and the angle C between them, you can find the third side (c) using the Law of Cosines:

c² = a² + b² – 2ab cos(C)

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

c = √(a² + b² – 2ab cos(C))

Step-by-Step Derivation/Explanation:

1. Square the lengths of the known sides: Calculate a² and b².
2. Multiply the known sides and 2: Calculate 2ab.
3. Find the cosine of the included angle: Calculate cos(C). Remember to convert angle C from degrees to radians if your cosine function requires it (radians = degrees * π / 180).
4. Multiply 2ab by cos(C): Calculate 2ab cos(C).
5. Subtract from the sum of squares: Calculate a² + b² – 2ab cos(C). This gives you c².
6. Take the square root: Calculate √c² to find the length of side c.

Variables Table

Variable	Meaning	Unit	Typical Range
a	Length of the first known side	Length (e.g., cm, m, inches)	> 0
b	Length of the second known side	Length (e.g., cm, m, inches)	> 0
C	Included angle between sides a and b	Degrees	0° < C < 180°
c	Length of the missing side opposite angle C	Length (e.g., cm, m, inches)	> 0
cos(C)	Cosine of angle C	Dimensionless	-1 to 1

Practical Examples (Real-World Use Cases)

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 60 degrees. They want to find the length of the third side.

• Side a = 120 m
• Side b = 150 m
• Angle C = 60°

Using the find missing side of triangle calculator (or the Law of Cosines):
c² = 120² + 150² – 2 * 120 * 150 * cos(60°)
c² = 14400 + 22500 – 36000 * 0.5
c² = 36900 – 18000 = 18900
c = √18900 ≈ 137.48 meters

The third side of the plot is approximately 137.48 meters long.

Example 2: Designing a Roof Truss

An engineer is designing a roof truss where two rafters are 5 meters and 7 meters long, and they meet at an angle of 110 degrees at the apex. They need to find the length of the base of the triangle formed (the third side).

• Side a = 5 m
• Side b = 7 m
• Angle C = 110°

Using the find missing side of triangle calculator:
cos(110°) ≈ -0.3420
c² = 5² + 7² – 2 * 5 * 7 * cos(110°)
c² = 25 + 49 – 70 * (-0.3420)
c² = 74 + 23.94 = 97.94
c = √97.94 ≈ 9.90 meters

The base of the triangle is approximately 9.90 meters long.

How to Use This Find Missing Side of Triangle Calculator

1. Enter Side a: Input the length of one of the known sides into the “Side a” field.
2. Enter Side b: Input the length of the other known side into the “Side b” 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 simply change the input values. The calculator will automatically update the results.
5. Read the Results:
   • Primary Result: The length of the missing side ‘c’ is displayed prominently.
   • Intermediate Values: You can also see values like Angle C in radians, cos(C), and c² to understand the calculation steps.
   • Formula Used: The Law of Cosines formula applied is shown.
6. View Chart and Table: The chart and table dynamically update to show how side ‘c’ changes with Angle C for the entered ‘a’ and ‘b’.
7. Reset: Click “Reset” to clear the inputs and results to their default values.
8. Copy Results: Click “Copy Results” to copy the main result, intermediates, and inputs to your clipboard.

This find missing side of triangle calculator is designed for ease of use, providing quick and accurate results based on the Law of Cosines.

Key Factors That Affect the Missing Side Calculation

1. Length of Side a: A larger value for side ‘a’ will generally lead to a larger value for side ‘c’, assuming ‘b’ and ‘C’ are constant and C < 90°.
2. Length of Side b: Similarly, a larger value for side ‘b’ will generally increase ‘c’, with ‘a’ and ‘C’ constant and C < 90°.
3. Magnitude of Angle C: This is crucial. As angle C increases from 0 towards 180 degrees:
   • If C < 90° (acute), cos(C) is positive, and the term -2ab cos(C) reduces c², making 'c' smaller than if C=90°.
   • If C = 90° (right), cos(90°)=0, and c² = a² + b² (Pythagorean theorem).
   • If C > 90° (obtuse), cos(C) is negative, and -2ab cos(C) becomes positive, increasing c², making ‘c’ larger than if C=90°.
4. Units of Sides a and b: The unit of side ‘c’ will be the same as the units used for sides ‘a’ and ‘b’. Consistency is key.
5. Accuracy of Angle Measurement: Small errors in measuring angle C can lead to noticeable differences in the calculated side ‘c’, especially when sides ‘a’ and ‘b’ are large.
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’, the input values of ‘a’, ‘b’, and ‘C’ (0 < C < 180) will always result in a valid 'c' that forms a triangle with 'a' and 'b'.

Understanding these factors helps in interpreting the results from the find missing side of triangle calculator and in real-world applications.

Frequently Asked Questions (FAQ)

1. What is the Law of Cosines?

The Law of Cosines relates the lengths of the sides of a triangle to the cosine of one of its angles. For a triangle with sides a, b, c, and angle C opposite side c, it states: c² = a² + b² – 2ab cos(C). Our find missing side of triangle calculator is based on this law.

2. When should I use the Law of Cosines instead of the Law of Sines?

Use the Law of Cosines when you know:

• Two sides and the included angle (SAS) – to find the third side.
• All three sides (SSS) – to find any angle.

Use the Law of Sines when you know:

• Two angles and any side (AAS or ASA) – to find other sides.
• Two sides and a non-included angle (SSA) – to find the angle opposite the other known side (but be aware of the ambiguous case).

3. Can this calculator find angles?

No, this specific find missing side of triangle calculator is designed to find a missing side when two sides and the included angle are known. To find angles given three sides, you’d rearrange the Law of Cosines formula or use a calculator designed for SSS.

4. What if my angle is 90 degrees?

If angle C is 90 degrees, cos(90°) = 0, and the Law of Cosines reduces to c² = a² + b², which is the Pythagorean theorem for right-angled triangles.

5. What units should I use for the sides?

You can use any unit of length (meters, feet, inches, etc.) for sides ‘a’ and ‘b’, as long as you are consistent. The calculated side ‘c’ will be in the same unit.

6. Does the calculator handle angles in radians?

This calculator expects the input angle C in degrees. It internally converts it to radians for the `Math.cos()` function.

7. What if the angle is 0 or 180 degrees?

The angle in a triangle must be greater than 0 and less than 180 degrees. The calculator validates this range. 0 or 180 degrees would form a degenerate triangle (a line).

8. Can I find the area using this information?

Yes, once you know two sides (a, b) and the included angle (C), the area of the triangle is (1/2) * a * b * sin(C). You can calculate this after finding ‘c’ or even before, using the initial inputs.

Related Tools and Internal Resources

Explore other calculators and resources that might be helpful:

• Triangle Area Calculator: Calculate the area of a triangle using various formulas, including SAS.
• Pythagorean Theorem Calculator: Specifically for right-angled triangles to find a missing side.
• Right Triangle Calculator: Solves for sides and angles of right triangles.
• Law of Sines Calculator: Use when you know AAS, ASA, or SSA to find missing sides or angles.
• Geometry Formulas: A collection of common geometry formulas.
• Math Calculators: Our main hub for various mathematical calculators.