2 Sides Of A Triangle Find The Third Calculator

Calculate Side ‘c’ using Law of Cosines

Enter the lengths of two sides (a and b) and the angle (C) between them to find the length of the third side (c).

How Side ‘c’ Changes with Angle ‘C’

What is a Find the Third Side of a Triangle Calculator?

A Find the Third Side of a Triangle Calculator, specifically one using the Law of Cosines, is a tool used to determine the length of the third side of a triangle when you know the lengths of the other two sides and the angle between them (the SAS or Side-Angle-Side case). This is particularly useful in geometry, trigonometry, engineering, and navigation where you have partial information about a triangle and need to find the missing side.

Anyone dealing with triangles, from students learning trigonometry to professionals in fields like surveying or physics, can use this Find the Third Side of a Triangle Calculator. It automates the Law of Cosines formula, making calculations quicker and less prone to manual error.

A common misconception is that you can find the third side with just two sides. While this is true for right-angled triangles using the Pythagorean theorem (if you know which sides are legs/hypotenuse), for a general triangle, you need more information – either the included angle (SAS) or other angles/sides to apply the Law of Sines or Cosines. This specific Find the Third Side of a Triangle Calculator focuses on the SAS case.

Find the Third Side of a Triangle Calculator: Formula and Mathematical Explanation

When you know two sides of a triangle (let’s call them ‘a’ and ‘b’) and the angle between them (angle ‘C’), you can find the length of the third side (‘c’) using the Law of Cosines.

The Law of Cosines formula is:

c² = a² + b² – 2ab cos(C)

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

c = √(a² + b² – 2ab cos(C))

Here’s a step-by-step breakdown:

1. Square the length of side ‘a’ (a²).
2. Square the length of side ‘b’ (b²).
3. Find the cosine of angle ‘C’ (cos(C)). Make sure angle C is in radians for most cosine functions, or your calculator handles degrees. Our Find the Third Side of a Triangle Calculator handles the degree-to-radian conversion.
4. Multiply 2 by ‘a’, ‘b’, and cos(C) (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)).
6. Take the square root of the result from step 5 to find the length of side ‘c’.

Variables in the Law of Cosines

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

If angle C is 90 degrees (a right angle), cos(90°) = 0, and the formula simplifies to c² = a² + b², which is the Pythagorean theorem. So, our Find the Third Side of a Triangle Calculator can also be used for right-angled triangles if you input 90 degrees for the angle.

Practical Examples (Real-World Use Cases)

Example 1: Surveying Land

A surveyor measures two sides of a triangular plot of land as 100 meters and 120 meters, with the angle between these sides being 75 degrees. They need to find the length of the third side.

• a = 100 m
• b = 120 m
• C = 75°

Using the Find the Third Side of a Triangle Calculator (or the formula c = √(100² + 120² – 2 * 100 * 120 * cos(75°))):

c ≈ √(10000 + 14400 – 24000 * 0.2588) ≈ √(24400 – 6211.2) ≈ √18188.8 ≈ 134.87 meters.

The third side is approximately 134.87 meters long.

Example 2: Navigation

A ship sails 15 nautical miles on one bearing, then changes course and sails 20 nautical miles. The angle between the two legs of the journey was 110 degrees. How far is the ship from its starting point?

• a = 15 nm
• b = 20 nm
• C = 110°

Using the Find the Third Side of a Triangle Calculator (c = √(15² + 20² – 2 * 15 * 20 * cos(110°))):

c ≈ √(225 + 400 – 600 * (-0.3420)) ≈ √(625 + 205.2) ≈ √830.2 ≈ 28.81 nautical miles.

The ship is approximately 28.81 nautical miles from its start.

How to Use This Find the Third Side of a Triangle Calculator

1. Enter Side a: Input the length of one of the known sides into the “Length of Side a” field.
2. Enter Side b: Input the length of the other known side 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. Ensure this angle is greater than 0 and less than 180 degrees. If you have a right-angled triangle and ‘a’ and ‘b’ are the legs, enter 90 degrees.
4. Calculate: Click the “Calculate” button. The Find the Third Side of a Triangle Calculator will instantly display the length of side c, intermediate values, and the formula used.
5. Read Results: The primary result is the length of side ‘c’. Intermediate values like Angle C in radians, a²+b², and 2ab cos(C) are also shown for clarity.
6. Reset: Click “Reset” to clear the fields and start over with default values.
7. Copy Results: Click “Copy Results” to copy the main result and intermediate values to your clipboard.

If you have a right-angled triangle and know one leg and the hypotenuse, you need to rearrange the Pythagorean theorem (a² + b² = c², where c is the hypotenuse) to find the other leg (e.g., b = √(c² – a²)). This calculator is primarily for the SAS case but can handle the right-angled case if legs are given (by setting angle C to 90).

Key Factors That Affect Find the Third Side of a Triangle Calculator Results

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.
   • As Angle C approaches 180 degrees (a straight line), side ‘c’ approaches the sum of ‘a’ and ‘b’ (c ≈ a + b).
   • As Angle C approaches 0 degrees (sides folding onto each other), side ‘c’ approaches the absolute difference between ‘a’ and ‘b’ (c ≈ |a – b|).
   • When Angle C is 90 degrees (right angle), cos(C)=0, and c = √(a² + b²).
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 Find the Third Side of a Triangle Calculator doesn’t convert units; it assumes consistency.
5. Accuracy of Input Values: Small errors in the input lengths or angle can lead to different results for side ‘c’, especially if the triangle is very thin or very wide.
6. Angle Unit (Degrees vs. Radians): Our Find the Third Side of a Triangle Calculator expects the angle in degrees and converts it internally. If you are doing manual calculations, be sure your cosine function uses the correct units.

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). Our Find the Third Side of a Triangle Calculator is based on this.

2. When should I use the Law of Cosines?

Use it when you know two sides and the included angle (SAS) to find the third side, or when you know all three sides (SSS) and want to find an angle.

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

Yes. If you know the two legs (a and b), enter their lengths and set Angle C to 90 degrees. The Find the Third Side of a Triangle Calculator will then use c² = a² + b² because cos(90°) = 0.

4. What if I know one leg and the hypotenuse of a right triangle?

This calculator is set up for SAS (two sides and included angle). For a right triangle where you know a leg and the hypotenuse (e.g., ‘a’ and ‘c’), you’d rearrange Pythagoras: b = √(c² – a²). You would need a different calculator or do it manually.

5. What if my angle is greater than 180 degrees or less than 0?

A triangle cannot have an internal angle greater than 180 or less than 0 degrees. The calculator expects an angle between 0 and 180 (exclusive of 0 and 180 for a non-degenerate triangle).

6. What units does the calculator use?

The Find the Third Side of a Triangle Calculator works with any consistent units of length (cm, meters, inches, feet, etc.) for sides a and b. The output for side c will be in the same units.

7. What is the triangle inequality theorem?

It states that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side (a + b > c, a + c > b, b + c > a). Our calculated ‘c’ will always satisfy this if 0 < C < 180.

8. How does the chart work?

The chart shows how the length of side ‘c’ changes as you vary Angle C from 1 to 179 degrees, keeping the entered values of ‘a’ and ‘b’ constant. You can see ‘c’ is smallest when C is small and largest when C is large.

Related Tools and Internal Resources

• Triangle Area Calculator: Calculate the area of a triangle using various formulas (base-height, Heron’s, SAS).
• Pythagorean Theorem Calculator: Specifically for right-angled triangles to find a missing side given two others.
• Law of Sines Calculator: Use when you know two angles and one side (AAS or ASA) or two sides and a non-included angle (SSA).
• Angle Conversion Calculator: Convert between degrees, radians, and other angle units.
• Right Triangle Calculator: Solves for all sides and angles of a right triangle.
• Geometry Calculators: A collection of various geometry-related calculators.