Find Side Of Triangle With 2 Sides Calculator

Triangle Side Calculator

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

What is the Find Side of Triangle with 2 Sides Calculator?

The find side of triangle with 2 sides calculator is a tool designed to calculate the length of the third side of any triangle when you know the lengths of the other two sides and the angle between them. This is typically done using the Law of Cosines, a fundamental theorem in trigonometry. You can also use it for right-angled triangles (where the angle is 90 degrees), in which case it simplifies to the Pythagorean theorem.

This calculator is useful for students, engineers, architects, and anyone dealing with geometry or needing to determine unknown dimensions of a triangle. It helps avoid manual calculations and provides quick, accurate results. Our find side of triangle with 2 sides calculator is particularly handy when you don’t have a right-angled triangle, where the Pythagorean theorem alone isn’t sufficient.

Common misconceptions include thinking you only need two sides without the included angle (which isn’t enough information for a general triangle) or that it only works for right-angled triangles. The Law of Cosines, which this find side of triangle with 2 sides calculator uses, is applicable to *any* triangle.

Find Side of Triangle with 2 Sides Calculator Formula and Mathematical Explanation

The primary formula used by the find side of triangle with 2 sides calculator is the Law of Cosines. If you have a triangle with sides a, b, and c, and the angles opposite these sides are A, B, and C respectively, the Law of Cosines states:

• c² = a² + b² – 2ab cos(C)
• a² = b² + c² – 2bc cos(A)
• b² = a² + c² – 2ac cos(B)

To find side c, given sides a, b, and the angle C between them, we rearrange the first formula:

c = √(a² + b² – 2ab cos(C))

Here’s a step-by-step derivation for finding side c:

1. Square the lengths of the known sides: a² and b².
2. Multiply the lengths of the known sides by 2: 2ab.
3. Find the cosine of the angle C between sides a and b. Note: The angle C must be in radians for most programming language cosine functions, so convert from degrees if necessary (radians = degrees * π / 180).
4. Multiply the result from step 2 by the result from step 3: 2ab cos(C).
5. Subtract the result from step 4 from the sum of the results from step 1: a² + b² – 2ab cos(C). This gives c².
6. Take the square root of the result from step 5 to find the length of side c.

Variables Table

Variable	Meaning	Unit	Typical Range
a	Length of the first known side	Length units (e.g., cm, m, inches)	> 0
b	Length of the second known side	Length units (e.g., cm, m, inches)	> 0
C	Angle between sides a and b	Degrees (for input), Radians (for calculation)	0 < C < 180 degrees
c	Length of the unknown side opposite angle C	Length units (e.g., cm, m, inches)	> 0
cos(C)	Cosine of angle C	Dimensionless	-1 to 1

Table explaining the variables used in the Law of Cosines.

Practical Examples (Real-World Use Cases)

Example 1: Surveying Land

A surveyor needs to find the distance across a lake (side c). They measure the distance from a point to one edge of the lake as 150 meters (side a) and to the other edge as 200 meters (side b). The angle between these two lines is measured as 60 degrees (angle C).

• Side a = 150 m
• Side b = 200 m
• Angle C = 60 degrees

Using the find side of triangle with 2 sides calculator or the formula c = √(150² + 200² – 2 * 150 * 200 * cos(60°)):

c = √(22500 + 40000 – 60000 * 0.5) = √(62500 – 30000) = √32500 ≈ 180.28 meters.

The distance across the lake is approximately 180.28 meters.

Example 2: Navigation

A boat travels 10 km east (side a), then turns 120 degrees and travels 15 km (side b). How far is the boat from its starting point (side c)? The angle *inside* the triangle formed by the start, turn, and end points is 180 – 120 = 60 degrees if the turn was external, but if we consider the angle *between* the 10km and 15km paths directly, it depends on how the 120-degree turn is defined. Let’s assume the angle *between* the 10km path and the 15km path is 180 – 120 = 60 degrees, or if the 120 was the internal angle C, then:

• Side a = 10 km
• Side b = 15 km
• Angle C = 120 degrees

Using the find side of triangle with 2 sides calculator or the formula c = √(10² + 15² – 2 * 10 * 15 * cos(120°)):

cos(120°) = -0.5

c = √(100 + 225 – 300 * (-0.5)) = √(325 + 150) = √475 ≈ 21.79 km.

The boat is approximately 21.79 km from its starting point. Using a find side of triangle with 2 sides calculator makes these calculations quick.

How to Use This Find Side of Triangle with 2 Sides 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 the angle is greater than 0 and less than 180 degrees.
4. View Results: The calculator will automatically update and display the length of side c, along with intermediate calculations like a², b², 2ab, cos(C), and c².
5. Interpret Results: The “Side c” value is the length of the third side of your triangle. The units will be the same as the units used for sides a and b.
6. Use the Chart: The chart below the calculator shows how the length of side ‘c’ would change if you varied the Angle C while keeping ‘a’ and ‘b’ constant.
7. Reset: Click “Reset” to return to the default values.
8. Copy: Click “Copy Results” to copy the main result and key values to your clipboard.

This find side of triangle with 2 sides calculator provides a clear output for side ‘c’.

Key Factors That Affect Find Side of Triangle with 2 Sides Calculator Results

1. Accuracy of Side a Measurement: Any error in measuring side a will directly propagate into the calculation of side c. More precise measurements lead to more accurate results from the find side of triangle with 2 sides calculator.
2. Accuracy of Side b Measurement: Similar to side a, the precision of the side b measurement is crucial for the accuracy of side c.
3. Accuracy of Angle C Measurement: The angle C is critical. Small errors in the angle, especially when the angle is close to 0 or 180 degrees, can lead to larger errors in the calculated side c. Using precise instruments for angle measurement is important.
4. Units Used: Ensure that both side a and side b are measured in the same units. The resulting side c will be in those same units. The find side of triangle with 2 sides calculator assumes consistent units.
5. 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’, if ‘c’ turns out to be very large or small, it might indicate the input values form an unusual triangle.
6. Rounding: The number of decimal places used in intermediate calculations and the final result can affect precision, though our find side of triangle with 2 sides calculator aims for good precision.

Frequently Asked Questions (FAQ)

1. What if I have two sides and an angle that is NOT between them?

If you have two sides and an angle not between them (e.g., sides a, b, and angle A), you would use the Law of Sines first to find another angle, then proceed. This calculator is specifically for when you have sides a, b, and the included angle C.

2. Can I use this find side of triangle with 2 sides calculator for a right-angled triangle?

Yes. If your triangle is right-angled and the angle C you enter is 90 degrees, cos(90°) = 0, so the formula c² = a² + b² – 2ab * 0 simplifies to c² = a² + b², which is the Pythagorean theorem. So, enter 90 degrees for angle C if ‘a’ and ‘b’ are the legs.

3. What units should I use for the sides?

You can use any units (cm, meters, inches, feet, etc.) for sides a and b, as long as you use the SAME units for both. The result for side c will be in those same units. The find side of triangle with 2 sides calculator is unit-agnostic in that sense.

4. Why does the angle C have to be between 0 and 180 degrees?

In any valid triangle, the sum of internal angles is 180 degrees, and each individual angle must be greater than 0 and less than 180 degrees. An angle of 0 or 180 would result in a degenerate triangle (a line).

5. What is the Law of Cosines?

The Law of Cosines is a theorem relating the lengths of the sides of a triangle to the cosine of one of its angles. It’s a generalization of the Pythagorean theorem and is used by this find side of triangle with 2 sides calculator.

6. Can I find the angles using this calculator?

No, this calculator is designed to find the length of a side given two sides and the included angle. You would need a different calculator or rearrange the Law of Cosines to solve for an angle if you have all three sides. Check our angle calculator.

7. What if my input values don’t form a valid triangle?

The formula will still produce a result for ‘c’ as long as a, b are positive and 0 < C < 180. However, the triangle inequality theorem (a+b>c, a+c>b, b+c>a) must be satisfied for a physical triangle to exist with these dimensions.

8. How accurate is this find side of triangle with 2 sides calculator?

The calculator uses standard mathematical formulas and is as accurate as the input values you provide. The internal calculations use high precision, but the final result is rounded for display.

Related Tools and Internal Resources

• Triangle Area Calculator: Calculate the area of a triangle using various formulas.
• Right Triangle Calculator: Specifically for solving right-angled triangles.
• Angle Calculator: Find angles of a triangle given sides.
• Pythagorean Theorem Calculator: For right-angled triangles to find sides or hypotenuse.
• Geometry Formulas: A collection of useful geometry formulas and explanations.
• Math Calculators: Explore other math-related calculators.