Find The Missing Side Of A Obtuse Triangle Calculator

Calculate the Missing Side

Enter two side lengths and the obtuse angle (greater than 90° and less than 180°) between them to find the third side of the obtuse triangle using the Law of Cosines.

What is an Obtuse Triangle Missing Side Calculator?

An obtuse triangle missing side calculator is a tool used to determine the length of an unknown side of an obtuse triangle when you have enough information about its other sides and angles. Specifically, it often uses the Law of Cosines if you know two sides and the included angle (which must be obtuse, i.e., greater than 90 degrees and less than 180 degrees) or the Law of Sines if other combinations of sides and angles are known, including the fact that one angle is obtuse. Our calculator focuses on the case where two sides and the obtuse angle between them are known, applying the Law of Cosines to find the side opposite the obtuse angle.

This calculator is particularly useful for students studying geometry and trigonometry, engineers, architects, and anyone needing to solve for triangle dimensions where one angle is greater than 90 degrees. It simplifies the process of applying the Law of Cosines.

Common misconceptions include thinking it can be used directly for right triangles (where the Pythagorean theorem is simpler) or acute triangles without adjustment, or that any set of three values will define an obtuse triangle.

Obtuse Triangle Missing Side Calculator Formula and Mathematical Explanation

When you know two sides of a triangle (let’s call them ‘a’ and ‘b’) and the angle ‘C’ between them, and angle ‘C’ is obtuse (90° < C < 180°), you can find the length of the side 'c' (opposite angle C) using the Law of Cosines:

c² = a² + b² – 2ab cos(C)

So, the length of side ‘c’ is:

c = √(a² + b² – 2ab cos(C))

Here’s a step-by-step breakdown:

1. Square the lengths of the two known sides (a² and b²).
2. Add these squares together (a² + b²).
3. Multiply the lengths of the two known sides by 2 (2ab).
4. Find the cosine of the obtuse angle C (cos(C)). Note: For an obtuse angle (90° < C < 180°), cos(C) will be negative.
5. Multiply the results from steps 3 and 4 (2ab cos(C)).
6. Subtract the result from step 5 from the result of step 2 (a² + b² – 2ab cos(C)). Since cos(C) is negative for an obtuse angle, this step effectively becomes an addition of a positive value.
7. Take the square root of the result from step 6 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	The obtuse angle between sides a and b	Degrees	90° < C < 180°
c	Length of the missing side opposite angle C	Length units (e.g., cm, m, inches)	> |a-b| and < a+b, and is the longest side
cos(C)	Cosine of angle C	Dimensionless	-1 < cos(C) < 0

Variables used in the obtuse triangle missing side calculator based on the Law of Cosines.

Practical Examples (Real-World Use Cases)

Let’s see how the obtuse triangle missing side calculator works with some examples.

Example 1: Surveying a Plot of Land

A surveyor measures two sides of a triangular plot of land as 150 meters and 100 meters. The angle between these two sides is measured to be 110°. What is the length of the third side?

• Side a = 150 m
• Side b = 100 m
• Angle C = 110°

Using the formula c = √(150² + 100² – 2 * 150 * 100 * cos(110°)):

c ≈ √(22500 + 10000 – 30000 * (-0.3420)) ≈ √(32500 + 10260) ≈ √42760 ≈ 206.78 meters.

The third side is approximately 206.78 meters.

Example 2: Engineering a Support Structure

An engineer is designing a support structure where two beams of length 8 feet and 5 feet meet at an angle of 135°. What is the distance between the other ends of the beams?

• Side a = 8 ft
• Side b = 5 ft
• Angle C = 135°

Using the formula c = √(8² + 5² – 2 * 8 * 5 * cos(135°)):

c ≈ √(64 + 25 – 80 * (-0.7071)) ≈ √(89 + 56.568) ≈ √145.568 ≈ 12.065 feet.

The distance is approximately 12.065 feet.

How to Use This Obtuse Triangle Missing Side 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 obtuse angle (between 90° and 180°) formed by sides a and b into the “Angle C (degrees)” field. Ensure the angle is correctly entered in degrees.
4. Calculate: Click the “Calculate Side c” button. The calculator will instantly display the length of the missing side ‘c’, along with intermediate calculations.
5. Read Results: The primary result is the length of side ‘c’. Intermediate results show the angle in radians, a²+b², and 2ab cos(C) for clarity.
6. Reset: Use the “Reset” button to clear the fields to their default values for a new calculation with the obtuse triangle missing side calculator.
7. Copy Results: Use the “Copy Results” button to copy the input values and the calculated results to your clipboard.

When using the obtuse triangle missing side calculator, make sure your angle is indeed obtuse and between the two sides you have entered.

Key Factors That Affect Obtuse Triangle Missing Side Calculation Results

Several factors influence the calculated length of the missing side in an obtuse triangle:

• Lengths of Sides a and b: Larger values for ‘a’ and ‘b’ will generally result in a larger value for ‘c’.
• Magnitude of Angle C: As the obtuse angle C gets closer to 180°, the value of cos(C) approaches -1, making the term -2ab cos(C) larger and positive, thus increasing the length of ‘c’. As C gets closer to 90° (but still obtuse), cos(C) is closer to 0, and ‘c’ will be closer to √(a²+b²).
• Accuracy of Input Values: Small errors in the measured sides or angle can lead to inaccuracies in the calculated side ‘c’, especially if the angle is very close to 90° or 180°.
• 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.
• Angle Being Obtuse: The formula and calculator are specifically applied here assuming C is obtuse, which means cos(C) is negative. This is crucial for the Law of Cosines calculation.
• Triangle Inequality: The calculated side ‘c’ must satisfy the triangle inequality theorem with sides ‘a’ and ‘b’ (a+b > c, a+c > b, b+c > a). In an obtuse triangle, the side opposite the obtuse angle (c in our case) is the longest side.

Frequently Asked Questions (FAQ)

Q1: What is an obtuse triangle?

A1: An obtuse triangle is a triangle in which one of the interior angles is greater than 90 degrees (an obtuse angle).

Q2: Can I use this calculator if the angle is not obtuse?

A2: The Law of Cosines formula c² = a² + b² – 2ab cos(C) works for any triangle (acute, right, or obtuse). However, this specific obtuse triangle missing side calculator is designed and explained with the assumption that angle C is obtuse (90° < C < 180°) as per the topic.

Q3: What if the angle entered is 90 degrees or less?

A3: If you enter 90 degrees, the formula simplifies to the Pythagorean theorem (c² = a² + b² because cos(90°)=0). If less than 90, it’s an acute angle, and the Law of Cosines still applies, but our calculator has validation for C > 90.

Q4: What if my angle is 180 degrees or more?

A4: A triangle cannot have an interior angle of 180 degrees or more, as the sum of angles must be 180 degrees. The calculator validates for C < 180.

Q5: Why is cos(C) negative for an obtuse angle C?

A5: In the unit circle, angles between 90° and 180° fall in the second quadrant, where the x-coordinate (which represents the cosine value) is negative.

Q6: What if I know two angles and one side, and one angle is obtuse?

A6: If you know two angles (A and B, one obtuse) and a side (say ‘a’), you can find the third angle (C = 180 – A – B) and then use the Law of Sines (b/sin(B) = a/sin(A), c/sin(C) = a/sin(A)) to find the other sides. This obtuse triangle missing side calculator is for the SAS case with an obtuse angle.

Q7: Is the side opposite the obtuse angle always the longest side?

A7: Yes, in any triangle, the longest side is always opposite the largest angle. Since the obtuse angle is the largest angle in an obtuse triangle, the side opposite it is the longest.

Q8: What happens if the input values don’t form a valid obtuse triangle with the given angle between the sides?

A8: The calculator requires sides to be positive and the angle to be between 90 and 180 exclusive. If these conditions are met, a valid triangle side ‘c’ can be calculated using the Law of Cosines for the SAS case.

Related Tools and Internal Resources

For other triangle-related calculations and geometry tools, check out:

Using the obtuse triangle missing side calculator alongside these resources can provide a comprehensive understanding of triangle calculations.