Missing Side of an Obtuse Triangle Calculator
Calculate the Missing Side
This calculator helps you find the missing side of an obtuse triangle when you know two sides and the obtuse angle opposite one of them.
What is a Missing Side of an Obtuse Triangle Calculator?
A missing side of an obtuse triangle calculator is a tool used to determine the length of an unknown side of a triangle that contains one angle greater than 90 degrees (an obtuse angle). When you have certain information about the triangle, such as the lengths of two sides and the measure of the obtuse angle opposite one of them, or two angles (one obtuse) and a side, this calculator can apply trigonometric principles like the Law of Sines and the Law of Cosines to find the missing side.
This calculator is specifically designed for scenarios where you know the triangle is obtuse and are given sides 'a' and 'b', and the obtuse angle 'A' opposite side 'a'. It helps find the length of side 'c'.
It's useful for students studying trigonometry, engineers, architects, and anyone dealing with geometric problems involving non-right-angled triangles, especially obtuse ones.
Common misconceptions include assuming the Pythagorean theorem applies (it only applies to right-angled triangles) or that the Law of Sines always gives a unique solution without considering the obtuse angle constraint (the ambiguous case).
Missing Side of an Obtuse Triangle Calculator Formula and Mathematical Explanation
When we know two sides (a and b) and the obtuse angle A (opposite side a) of a triangle, we can find the third side (c) using the Law of Sines and the fact that the sum of angles in a triangle is 180 degrees.
The steps are:
- Apply the Law of Sines to find Angle B:
sin(B) / b = sin(A) / a
sin(B) = (b * sin(A)) / a
Since A is obtuse, B must be acute, so B = arcsin((b * sin(A)) / a). We must check if (b * sin(A)) / a is between 0 and 1 for a valid triangle. - Calculate Angle C:
C = 180° – A – B - Apply the Law of Sines again to find side c:
c / sin(C) = a / sin(A)
c = (a * sin(C)) / sin(A)
The missing side of an obtuse triangle calculator implements these formulas.
Variables Table
| Variable | Meaning | Unit | Typical Range |
|---|---|---|---|
| a | Length of side opposite angle A | Length units (e.g., m, cm, ft) | > 0 |
| b | Length of another side | Length units (e.g., m, cm, ft) | > 0 |
| c | Length of the missing side | Length units (e.g., m, cm, ft) | > 0 (calculated) |
| A | Obtuse angle opposite side a | Degrees | 90° < A < 180° |
| B | Angle opposite side b | Degrees | 0° < B < 90° (calculated) |
| C | Third angle | Degrees | 0° < C < (180°-A) (calculated) |
Practical Examples (Real-World Use Cases)
Example 1: Land Surveying
A surveyor measures two sides of a triangular plot of land as 120 meters (side a) and 80 meters (side b). They also measure the angle opposite the 120m side as 110° (angle A). They need to find the length of the third side (c).
Inputs: a = 120, b = 80, A = 110°
Using the missing side of an obtuse triangle calculator:
- sin(B) = (80 * sin(110°)) / 120 ≈ 0.6264
- B ≈ arcsin(0.6264) ≈ 38.79°
- C ≈ 180° – 110° – 38.79° ≈ 31.21°
- c ≈ (120 * sin(31.21°)) / sin(110°) ≈ 66.05 meters
The third side of the plot is approximately 66.05 meters.
Example 2: Navigation
A boat travels from point P. It travels 15 nautical miles to point Q (side b), then turns and travels to point R (side c). The angle at P (angle between PQ extended and PR) is not directly known, but the angle at Q (angle PQR, which is obtuse) is 130° (let's rephrase for our calculator: if angle at Q is B=130, our A must be different. Let's assume angle opposite a known side is obtuse).
Let's say a ship sails 10 nautical miles (side a), turns, and the angle opposite this leg is 125 degrees (angle A). Another leg from the start is 7 nautical miles (side b). What is the distance between the end points (side c)?
Inputs: a = 10, b = 7, A = 125°
Using the missing side of an obtuse triangle calculator:
- sin(B) = (7 * sin(125°)) / 10 ≈ 0.5734
- B ≈ arcsin(0.5734) ≈ 34.99°
- C ≈ 180° – 125° – 34.99° ≈ 20.01°
- c ≈ (10 * sin(20.01°)) / sin(125°) ≈ 4.17 nautical miles
The distance between the end points is about 4.17 nautical miles.
How to Use This Missing Side of an Obtuse Triangle Calculator
- Enter Side 'a': Input the length of the side opposite the obtuse angle you know.
- Enter Side 'b': Input the length of one of the other sides.
- Enter Obtuse Angle 'A': Input the measure of the obtuse angle (between 90 and 180 degrees) that is opposite side 'a'.
- Calculate: The calculator automatically updates, or you can click "Calculate".
- Read Results: The calculator will display the length of the missing side 'c', and the other two angles B and C. It will also indicate if a valid triangle can be formed. The primary result is side 'c'.
- Use the Chart: The bar chart visually represents the lengths of the three sides of the triangle.
Decision-making: If the calculator indicates "Triangle cannot be formed," re-check your input values. The side opposite the obtuse angle (a) must be the largest side if side b and angle A are such that b*sin(A)/a > 1, or angle C becomes zero or negative.
Key Factors That Affect Missing Side of an Obtuse Triangle Calculator Results
- Accuracy of Input Values: Small errors in measuring sides or the obtuse angle can lead to larger errors in the calculated side and angles.
- The Obtuse Angle Value: The angle A being greater than 90 degrees is crucial for the context of this calculator. It influences the possible range of values for other angles and sides.
- Ratio of Sides (b/a) and sin(A): The value of (b * sin(A)) / a determines if a triangle is possible and the value of angle B. If it's greater than 1, no such triangle exists.
- Sum of Angles: The fact that A + B + C must equal 180 degrees is fundamental. If the calculated angles don't allow for this, the input values might be incorrect for an obtuse triangle of this configuration.
- Unit Consistency: Ensure all side lengths are in the same units. The calculator doesn't convert units.
- Rounding: Rounding intermediate values during manual calculation can affect the final result. The calculator uses higher precision internally.
Frequently Asked Questions (FAQ)
Q1: What is an obtuse triangle?
A: An obtuse triangle is a triangle in which one of the angles is greater than 90 degrees (an obtuse angle).
Q2: Can a triangle have more than one obtuse angle?
A: No, because the sum of angles in any triangle is 180 degrees. If two angles were obtuse (each > 90°), their sum alone would exceed 180°, which is impossible.
Q3: What formula is used by this missing side of an obtuse triangle calculator?
A: It primarily uses the Law of Sines (a/sin A = b/sin B = c/sin C) and the property that angles sum to 180°.
Q4: Why does the calculator say "Triangle cannot be formed"?
A: This happens if the given side 'b', angle 'A', and side 'a' values are such that (b * sin(A)) / a > 1, meaning no real angle B exists, or if the calculated angle C is zero or negative.
Q5: Can I use this calculator for a right or acute triangle?
A: This specific calculator is designed assuming angle A is obtuse (90° < A < 180°). For right or acute triangles, you might need a more general triangle solver or a specific right triangle calculator.
Q6: What if I know two sides and the angle between them, and I know one of the other angles is obtuse?
A: If you know sides 'a' and 'b' and the included angle 'C', you'd first find side 'c' using the Law of Cosines (c² = a² + b² – 2ab cos(C)). Then you find the other angles using the Law of Sines or Cosines and check which is obtuse. This calculator assumes you know the obtuse angle A opposite side a.
Q7: How do I know which angle is the obtuse one?
A: For this calculator, you must input the obtuse angle as 'A' and the side opposite it as 'a'. In a general problem, the side opposite the obtuse angle is the longest side of the triangle, but that's not always enough to identify it without more info.
Q8: What units should I use for the sides?
A: You can use any units (meters, feet, cm, etc.), but be consistent for all sides. The output for the missing side will be in the same unit.
Related Tools and Internal Resources
- Right Triangle Calculator: For triangles with a 90-degree angle.
- Triangle Area Calculator: Calculate the area given various inputs.
- Law of Sines Calculator: Solves triangles using the Law of Sines.
- Law of Cosines Calculator: Solves triangles using the Law of Cosines.
- Geometry Calculators: A collection of calculators for various geometric shapes.
- Angle Converter: Convert between different angle units (degrees, radians).
These resources, including the missing side of an obtuse triangle calculator, can help with various geometry and trigonometry problems. Explore our geometry calculator section for more tools.