Find Third Length Of Triangle Calculator

Easily calculate the missing side of any triangle (SAS or right-angled) with our Find Third Length of Triangle Calculator.

Triangle Calculator

Triangle Visualization

What is a Find Third Length of Triangle Calculator?

A find third length of triangle calculator is a specialized tool designed to determine the length of one side of a triangle when you know the lengths of the other two sides and the angle between them (SAS – Side-Angle-Side), or when you have a right-angled triangle and know two of its sides. It primarily uses the Law of Cosines for general triangles and the Pythagorean theorem for right-angled triangles to calculate the missing side length. This calculator is invaluable for students, engineers, architects, and anyone working with geometric figures.

It helps solve problems where you can't directly measure a side but have other relevant information. For instance, if you know the lengths of two sides of a piece of land and the angle where they meet, you can use the find third length of triangle calculator (using the Law of Cosines) to find the length of the third side. Similarly, if you know the lengths of the two shorter sides of a right-angled support brace, you can find the length of the hypotenuse.

Common misconceptions include thinking it can find a side with only angles or only one side (unless it's a special triangle with more info). You typically need at least two sides and an included angle for a general triangle, or two sides for a right triangle, to find the third side using these methods.

Find Third Length of Triangle Calculator: Formulas and Mathematical Explanation

The core formulas used by the find third length of triangle calculator depend on the type of triangle and the information provided:

1. Law of Cosines (for SAS – Side-Angle-Side)

When you know two sides (a and b) and the included angle (C), the length of the third side (c) is found using the Law of Cosines:

c² = a² + b² – 2ab * cos(C)

So, c = √(a² + b² – 2ab * cos(C))

Where 'a' and 'b' are the lengths of the known sides, and 'C' is the angle (in radians for the `cos` function) between them.

2. Pythagorean Theorem (for Right-Angled Triangles)

If the triangle is right-angled (one angle is 90 degrees), and you know two sides, you can find the third:

• If you know the two legs (a and b), the hypotenuse (c) is: c² = a² + b² => c = √(a² + b²)
• If you know one leg (a) and the hypotenuse (c), the other leg (b) is: b² = c² – a² => b = √(c² – a²)

3. Law of Sines (to find other angles after finding the third side)

Once the third side 'c' is found (using Law of Cosines), we can find the other angles (A and B) using the Law of Sines:

a/sin(A) = b/sin(B) = c/sin(C)

So, sin(A) = a * sin(C) / c, and sin(B) = b * sin(C) / c. Then A = arcsin(a * sin(C) / c) and B = arcsin(b * sin(C) / c).

Variables Table:

Variable	Meaning	Unit	Typical Range
a, b, c	Lengths of the sides of the triangle	(e.g., cm, m, inches)	> 0
A, B, C	Angles opposite sides a, b, c respectively	Degrees (for input), Radians (in calculation)	0° < Angle < 180° (Sum = 180°)

Table 1: Variables used in triangle calculations.

Practical Examples (Real-World Use Cases)

Example 1: Using Law of Cosines (SAS)

Imagine you are surveying a triangular piece of land. You measure two sides to be 100 meters and 120 meters, and the angle between these two sides is 75 degrees. You want to find the length of the third side.

• Side a = 100 m
• Side b = 120 m
• Angle C = 75°

Using the find third length of triangle calculator (or the Law of Cosines: c² = 100² + 120² – 2 * 100 * 120 * cos(75°)):

c² = 10000 + 14400 – 24000 * cos(75°) ≈ 24400 – 24000 * 0.2588 ≈ 24400 – 6211.2 = 18188.8

c ≈ √18188.8 ≈ 134.87 meters. The third side is approximately 134.87 meters long.

Example 2: Using Pythagorean Theorem

A carpenter is building a right-angled frame. One leg is 3 feet long, and the other leg is 4 feet long. They need to find the length of the hypotenuse.

• Leg a = 3 ft
• Leg b = 4 ft

Using the Pythagorean theorem (c² = a² + b²):

c² = 3² + 4² = 9 + 16 = 25

c = √25 = 5 feet. The hypotenuse is 5 feet long.

How to Use This Find Third Length of Triangle Calculator

1. Select Triangle Type: Choose between "General Triangle (SAS)" if you know two sides and the included angle, or "Right-Angled Triangle" if your triangle has a 90-degree angle.
2. Enter Known Values:
   • For SAS: Input the lengths of side 'a', side 'b', and the measure of angle 'C' in degrees.
   • For Right-Angled: Select if you're finding the hypotenuse (and input the two legs) or finding a leg (and input one leg and the hypotenuse).
3. Input Values: Carefully enter the known lengths and angles into the respective fields. Ensure angles are in degrees.
4. Calculate: Click the "Calculate" button (or the results will update automatically as you type if implemented that way).
5. Read Results: The calculator will display:
   • The length of the unknown third side (the primary result).
   • Other calculated values like the area of the triangle and the other two angles (if SAS was used or enough info for right-angled).
   • The formula used for the calculation.
6. Visualize: If available, look at the triangle visualization to get a graphical representation based on your inputs.
7. Reset: Click "Reset" to clear the fields and start a new calculation.

This find third length of triangle calculator is a powerful tool for quickly solving for an unknown side.

Key Factors That Affect Find Third Length of Triangle Calculator Results

1. Accuracy of Input Lengths: Small errors in measuring the known sides will directly impact the calculated length of the third side.
2. Accuracy of Input Angle: For the SAS case, the precision of the angle measurement is crucial. An error in the angle C affects the cos(C) value and thus the calculated side 'c'.
3. Triangle Type Selection: Using the Pythagorean theorem for a non-right-angled triangle (or Law of Cosines without the correct angle) will give incorrect results. Ensure you select the right mode.
4. Units Consistency: Ensure all length inputs are in the same units (e.g., all meters or all feet). The output will be in the same unit.
5. Angle Units: Our calculator takes angles in degrees, but the `cos` function in JavaScript uses radians. The conversion is handled internally, but it's important the input is in degrees.
6. Valid Triangle Conditions: For SAS, the angle C must be between 0 and 180 degrees (exclusive). For right-angled triangles finding a leg, the hypotenuse must be longer than the known leg.

Frequently Asked Questions (FAQ)

1. Can I use this calculator if I only know the angles?

No. Knowing only angles determines the shape of the triangle but not its size. You need at least one side length along with angles to find other side lengths (using the Law of Sines, but not directly with this SAS/Right-angled focus).

2. What if I have three sides and want to find the angles?

This find third length of triangle calculator is for finding a side. If you have three sides (SSS), you would use the Law of Cosines rearranged to find the angles.

3. What does SAS mean?

SAS stands for Side-Angle-Side. It means you know the lengths of two sides and the measure of the angle between those two sides.

4. Why is the angle input in degrees?

Degrees are more commonly used in everyday geometry problems than radians. The calculator converts degrees to radians internally for the trigonometric functions.

5. Can the calculator handle any values for sides and angles?

Side lengths must be positive numbers. For SAS, the angle must be between 0 and 180 degrees. For right-angled triangles, the hypotenuse must be the longest side.

6. 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).

7. What is the Pythagorean Theorem?

The Pythagorean Theorem applies only to right-angled triangles and states that the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides (legs): a² + b² = c².

8. How accurate is this find third length of triangle calculator?

The calculator uses standard mathematical formulas and is as accurate as the input values provided. Rounding may occur in the final displayed results.