Find Side Of A Triangle With Only One Angle Calculator

Calculate the third side of a triangle using the Law of Cosines when two sides and the included angle are known.

Triangle Side Calculator (SAS)

What is a Calculator to Find Side of Triangle with One Angle?

A “find side of triangle with one angle calculator” helps you determine the length of an unknown side of a triangle when you know the lengths of the other two sides and the angle between them (the included angle). This scenario is often referred to as the Side-Angle-Side (SAS) case in trigonometry. The primary mathematical tool used is the Law of Cosines.

While you can’t find all sides with *just* one angle and nothing else, if you have one angle and two sides, specifically the two sides that form that angle, you can find the third side. This calculator is designed for that SAS scenario. If you have one side and two angles (ASA or AAS), you would use the Law of Sines, which is a different principle.

This tool is useful for students studying trigonometry, engineers, architects, and anyone needing to solve for triangle dimensions without a right angle.

Common misconceptions include thinking you can solve a triangle with only one angle and no sides, or with one angle and only one side (which isn’t enough information to uniquely define a triangle).

Find Side of Triangle (SAS) Formula and Mathematical Explanation

To find the length of the third side (‘c’) when you know two sides (‘a’ and ‘b’) and the included angle (‘C’), we use the Law of Cosines:

c² = a² + b² – 2ab * cos(C)

Where:

• ‘a’ and ‘b’ are the lengths of the known sides.
• ‘C’ is the measure of the angle between sides ‘a’ and ‘b’, given in degrees.
• ‘cos(C)’ is the cosine of angle ‘C’. Note that the angle ‘C’ must be converted to radians before calculating its cosine if your calculator/function expects radians. (Degrees * π / 180 = Radians).
• ‘c’ is the length of the side opposite angle ‘C’, which we want to find.

The steps are:

1. Square the lengths of sides ‘a’ and ‘b’ (a², b²).
2. Convert angle ‘C’ from degrees to radians.
3. Calculate the cosine of angle ‘C’ (in radians).
4. Multiply 2 * a * b * cos(C).
5. Subtract the result from step 4 from the sum of a² + b². This gives c².
6. Take the square root of c² to find the length of side ‘c’.

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

Variables used in the Law of Cosines (SAS case).

Practical Examples (Real-World Use Cases)

Let’s see how to find the side of a triangle with one angle and two sides in practice.

Example 1: Surveying Land

A surveyor measures two sides of a triangular plot of land as 150 meters and 200 meters. The angle between these two sides is measured as 50 degrees. They want to find the length of the third side.

• Side a = 150 m
• Side b = 200 m
• Angle C = 50 degrees

Using the formula c² = a² + b² – 2ab * cos(C):

c² = 150² + 200² – 2 * 150 * 200 * cos(50°)

c² = 22500 + 40000 – 60000 * 0.6427876

c² = 62500 – 38567.256

c² = 23932.744

c = √23932.744 ≈ 154.70 meters

The third side is approximately 154.70 meters long.

Example 2: Navigation

A ship sails 30 miles East, then turns 110 degrees North of East and sails for 40 miles. How far is the ship from its starting point?

• Side a = 30 miles
• Side b = 40 miles
• The angle between these paths *inside* the triangle formed is 180° – 110° = 70 degrees. So, Angle C = 70 degrees.

Using the formula c² = a² + b² – 2ab * cos(C):

c² = 30² + 40² – 2 * 30 * 40 * cos(70°)

c² = 900 + 1600 – 2400 * 0.34202

c² = 2500 – 820.848

c² = 1679.152

c = √1679.152 ≈ 40.98 miles

The ship is approximately 40.98 miles from its starting point.

How to Use This Find Side of 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 measure of the angle between sides ‘a’ and ‘b’ (the included angle) in degrees into the “Included Angle ‘C'” field. The angle must be between 0 and 180 degrees exclusively.
4. Calculate: The calculator automatically updates the results as you type. You can also click the “Calculate Side ‘c'” button.
5. Read Results: The primary result is the length of side ‘c’. Intermediate values like the angle in radians, cos(C), a², b², and c² are also shown.
6. Reset: Click “Reset” to return to the default values.
7. Copy: Click “Copy Results” to copy the calculated side ‘c’ and intermediate values to your clipboard.

The chart below the calculator visually represents how side ‘c’ changes if you were to vary Angle C while keeping ‘a’ and ‘b’ constant.

Key Factors That Affect the Result

The length of the unknown side ‘c’ is influenced by:

1. Length of Side ‘a’: A longer side ‘a’ will generally result in a longer side ‘c’, especially if angle C is large.
2. Length of Side ‘b’: Similar to side ‘a’, a longer side ‘b’ typically leads to a longer side ‘c’.
3. Magnitude of Angle ‘C’: This is crucial.
   • As Angle ‘C’ approaches 180 degrees (a nearly flat triangle), side ‘c’ approaches the sum of ‘a’ and ‘b’.
   • As Angle ‘C’ approaches 0 degrees (a very thin triangle), side ‘c’ approaches the absolute difference between ‘a’ and ‘b’.
   • When Angle C is 90 degrees (right triangle), the Law of Cosines simplifies to the Pythagorean theorem (c² = a² + b² because cos(90°)=0).
4. Units of Sides ‘a’ and ‘b’: The unit of side ‘c’ will be the same as the units used for ‘a’ and ‘b’. Consistency is key.
5. Accuracy of Input Values: Small errors in the measured sides or angle can lead to errors in the calculated side ‘c’, especially when the angle is very small or close to 180 degrees.
6. Trigonometric Function (Cosine): The value of cos(C) directly impacts the term subtracted from a² + b².

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. It’s a generalization of the Pythagorean theorem and is used for non-right triangles. The formula to find a side is c² = a² + b² – 2ab * cos(C).

2. Can I use this calculator if I have one side and two angles (ASA or AAS)?

No, this calculator is specifically for the SAS (Side-Angle-Side) case using the Law of Cosines. If you have one side and two angles, you need to use the Law of Sines after finding the third angle (angles sum to 180°). You’d use a triangle solver that handles ASA/AAS.

3. What if my angle is 90 degrees?

If your angle C is 90 degrees, cos(90°) = 0, so the Law of Cosines becomes c² = a² + b² – 0, which is the Pythagorean theorem (c² = a² + b²). The calculator will still work correctly.

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

An internal angle of a triangle must be greater than 0 and less than 180 degrees. If it were 0 or 180, you would have a straight line, not a triangle.

5. Can I find the angles using this calculator?

No, this calculator finds a side given two sides and the included angle. To find angles, you would rearrange the Law of Cosines formula or use the Law of Sines if you have enough information. See our triangle angle calculator.

6. What units should I use?

You can use any unit for the lengths (meters, feet, inches, etc.), but be consistent. The output for side ‘c’ will be in the same unit as sides ‘a’ and ‘b’.

7. How accurate is the calculation?

The calculation is as accurate as the input values and the precision of the cosine function used. Our calculator uses standard JavaScript Math functions for high precision.

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

You would use a rearranged version of the Law of Cosines: cos(C) = (a² + b² – c²) / 2ab. You’d need a calculator designed for SSS (Side-Side-Side) to find angles.

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