Triangle Angle with 2 Sides (and More) Calculator
Use this calculator to find the angles and sides of a triangle given three sides (SSS) or two sides and the included angle (SAS). Our triangle angle with 2 sides calculator utilizes the Law of Cosines and Law of Sines for accurate results.
Triangle Calculator
What is a Triangle Angle with 2 Sides Calculator?
A triangle angle with 2 sides calculator is a tool designed to determine the unknown angles (and sometimes the third side) of a triangle when you know the lengths of two sides and either the third side (SSS – Side-Side-Side case) or the angle between the two known sides (SAS – Side-Angle-Side case). It primarily uses the Law of Cosines and the Law of Sines for calculations.
This calculator is particularly useful for students, engineers, architects, and anyone dealing with geometry or trigonometry problems where direct angle measurement is not possible but side lengths are known or measurable. By inputting two sides and either the third side or the included angle, you can find all three angles of the triangle.
Common misconceptions include thinking you can uniquely determine all angles with just two sides and *any* angle (the SSA case can be ambiguous) or with just two sides and no other information. Our triangle angle with 2 sides calculator handles the clear SSS and SAS cases.
Triangle Angle Calculator Formula and Mathematical Explanation
To find the angles of a triangle, especially when dealing with a triangle angle with 2 sides calculator scenario (like SSS or SAS), we rely on fundamental trigonometric laws:
1. Law of Cosines:
The Law of Cosines relates the lengths of the sides of a triangle to the cosine of one of its angles.
c² = a² + b² - 2ab * cos(C)b² = a² + c² - 2ac * cos(B)a² = b² + c² - 2bc * cos(A)
From these, we can find the angles if all three sides (a, b, c) are known (SSS case):
Angle A = arccos((b² + c² - a²) / (2bc))Angle B = arccos((a² + c² - b²) / (2ac))Angle C = arccos((a² + b² - c²) / (2ab))
In the SAS case (given a, b, and angle C), we first find side c using c² = a² + b² - 2ab * cos(C), then use the Law of Cosines or Sines to find angles A and B.
2. Law of Sines:
The Law of Sines relates the lengths of the sides of a triangle to the sines of its opposite angles:
a / sin(A) = b / sin(B) = c / sin(C)
After finding one angle and all sides (or the third side in SAS), we can use the Law of Sines to find the remaining angles. For example, after finding side 'c' and having angle 'C' in SAS, we can find angle A: sin(A) = (a * sin(C)) / c.
Triangle Validity Check:
For three lengths a, b, and c to form a triangle, the sum of any two sides must be greater than the third side (a+b > c, a+c > b, b+c > a). Also, angles must sum to 180 degrees, and no angle can be 0 or 180 degrees in a valid triangle formed from SAS.
Variables Table:
| Variable | Meaning | Unit | Typical Range |
|---|---|---|---|
| a, b, c | Lengths of the sides of the triangle | Units of length (e.g., m, cm, inches) | > 0 |
| A, B, C | Angles opposite to sides a, b, c respectively | Degrees or Radians | 0° < Angle < 180° (0 < Angle < π) |
Practical Examples (Real-World Use Cases)
Example 1: SSS Case (Surveying)
A surveyor measures three sides of a triangular plot of land as a = 50 meters, b = 70 meters, and c = 80 meters.
Inputs: Side a = 50, Side b = 70, Side c = 80
Using the Law of Cosines:
- Angle A = arccos((70² + 80² – 50²) / (2 * 70 * 80)) ≈ 38.21°
- Angle B = arccos((50² + 80² – 70²) / (2 * 50 * 80)) ≈ 60.00°
- Angle C = arccos((50² + 70² – 80²) / (2 * 50 * 70)) ≈ 81.79°
Outputs: Angle A ≈ 38.21°, Angle B ≈ 60.00°, Angle C ≈ 81.79°. The plot is a valid triangle.
Example 2: SAS Case (Navigation)
A ship sails 10 km on a bearing, then changes direction and sails 12 km. The angle between the two paths was 110 degrees.
Inputs: Side a = 10 km, Side b = 12 km, Angle C = 110°
First, find side c (distance between start and end):
c² = 10² + 12² - 2 * 10 * 12 * cos(110°) ≈ 100 + 144 - 240 * (-0.342) ≈ 244 + 82.08 ≈ 326.08
c ≈ sqrt(326.08) ≈ 18.06 km
Then find Angle A using Law of Sines: sin(A) = (10 * sin(110°)) / 18.06 ≈ (10 * 0.9397) / 18.06 ≈ 0.5203
Angle A ≈ arcsin(0.5203) ≈ 31.35°
Angle B = 180° - 110° - 31.35° ≈ 38.65°
Outputs: Side c ≈ 18.06 km, Angle A ≈ 31.35°, Angle B ≈ 38.65°.
How to Use This Triangle Angle with 2 Sides Calculator
- Select Mode: Choose "SSS (3 Sides)" if you know all three side lengths, or "SAS (2 Sides + Included Angle)" if you know two sides and the angle between them.
- Enter Values:
- For SSS: Input the lengths of sides a, b, and c.
- For SAS: Input the lengths of sides a and b, and the measure of angle C (in degrees) between them.
- Calculate: The calculator will automatically update as you type, or you can click "Calculate".
- Read Results: The calculator will display:
- The calculated angles A, B, and C (in degrees).
- Side c if you used SAS mode.
- Whether a valid triangle is formed.
- The type of triangle (e.g., scalene, isosceles, equilateral, right, acute, obtuse).
- The area of the triangle.
- A summary table and a bar chart of the angles.
- Decision-Making: Use the angles and side lengths for your specific application, whether it's surveying land, navigating, or solving a geometry problem. Our triangle angle with 2 sides calculator gives you the core geometric data.
Key Factors That Affect Triangle Angle Results
- Side Lengths (a, b, c): The relative lengths of the sides directly determine the angles. Small changes in side lengths can lead to significant changes in angles, especially in triangles with very different side lengths. The triangle inequality theorem (a+b>c, etc.) must be satisfied.
- Included Angle (C in SAS): In the SAS case, the angle between the two known sides is crucial. A larger included angle will result in a longer third side and different opposite angles compared to a smaller included angle with the same side lengths.
- Measurement Precision: The accuracy of your input values (side lengths and angles) directly impacts the accuracy of the calculated angles. More precise inputs give more precise results.
- Units: Ensure all side lengths are in the same units, and the angle is in degrees as expected by the calculator. Mixing units will lead to incorrect results.
- Rounding: The number of decimal places used in calculations and displayed in results affects precision. Our triangle angle with 2 sides calculator aims for reasonable precision.
- Calculator Mode (SSS vs. SAS): Using the correct mode based on the information you have is fundamental. SSS requires three sides, SAS requires two sides and the angle *between* them.
Frequently Asked Questions (FAQ)
- 1. What if I only know two sides and an angle NOT between them (SSA)?
- This calculator is primarily for SSS and SAS. The SSA case (two sides and a non-included angle) is known as the Ambiguous Case because it can result in zero, one, or two possible triangles. You would use the Law of Sines, but be mindful of multiple solutions. We might add an SSA calculator later.
- 2. Can I use this calculator for a right-angled triangle?
- Yes. If you input sides that form a right-angled triangle (e.g., a=3, b=4, c=5 for SSS), one of the calculated angles will be 90 degrees. For SAS, if you input C=90 degrees, it will calculate a right triangle.
- 3. What does "Invalid Triangle" mean?
- It means the given side lengths (in SSS) do not satisfy the triangle inequality theorem (the sum of any two sides must be greater than the third), or the angle in SAS is 0 or 180 or more. No triangle can be formed with those dimensions.
- 4. How is the area calculated?
- For SSS, we use Heron's formula. For SAS, we use Area = 0.5 * a * b * sin(C).
- 5. What is the difference between Law of Cosines and Law of Sines?
- The Law of Cosines relates all three sides to one angle, while the Law of Sines relates sides to the sines of their opposite angles. Cosines is used for SSS and finding the third side in SAS; Sines is often used after Cosines to find other angles more easily in SAS, or for SSA.
- 6. Why does my angle sum not equal exactly 180 degrees?
- It might be due to rounding in the displayed values. The internal calculations are more precise, but the sum of the displayed rounded angles might be very slightly off 180 (e.g., 179.99 or 180.01).
- 7. Can I enter side lengths in different units?
- No, all side lengths must be in the same unit (e.g., all meters or all inches). The triangle angle with 2 sides calculator does not convert units.
- 8. What if my included angle in SAS is 90 degrees?
- The calculator will correctly use cos(90°)=0, and the Law of Cosines simplifies to the Pythagorean theorem (c² = a² + b²).
Related Tools and Internal Resources
- Pythagorean Theorem Calculator: Useful for right-angled triangles specifically.
- Area of a Triangle Calculator: Calculates area using various formulas.
- Right Triangle Calculator: Solves right triangles given different inputs.
- Law of Sines Calculator: Focuses on the Law of Sines and the SSA case.
- Law of Cosines Calculator: Detailed calculator for Law of Cosines applications.
- Geometry Formulas Guide: A reference for various geometric formulas.