What is a Triangle Calculator?

A Triangle Calculator is a powerful geometry tool used to solve for the unknown sides, angles, perimeter, and area of a triangle, given a sufficient amount of information. Triangles are fundamental geometric shapes, and calculating their properties is essential in fields like engineering, physics, surveying, navigation, graphic design, and mathematics education.

Unlike basic calculators that might only handle right triangles or require all three sides, this versatile tool can solve triangles based on different combinations of known values, such as:

• SSS (Side-Side-Side): All three side lengths are known.
• SAS (Side-Angle-Side): Two sides and the angle *between* them are known.
• ASA (Angle-Side-Angle): Two angles and the side *between* them are known.
• AAS (Angle-Angle-Side): Two angles and a side *not* between them are known.
• SSA (Side-Side-Angle): Two sides and an angle *not* between them are known (Note: This case can sometimes be ambiguous, potentially leading to two valid triangles. This calculator attempts to find one valid solution).

How Does This Calculator Work?

This calculator uses trigonometric laws and geometric principles to solve the triangle based on the inputs you provide:

• Input: Enter exactly three known values into the corresponding fields (e.g., side 'a', side 'b', and angle 'C', or angle 'A', angle 'B', side 'c'). At least one input must be a side length. Select the correct angle unit (Degrees or Radians).
• Validation: The calculator first checks if exactly three valid inputs are provided, including at least one side. It also performs basic checks (e.g., sides must be positive, angles must be within valid ranges, the sum of two input angles cannot exceed 180°/π radians).
• Case Detection: It analyzes which combination of inputs you provided (SSS, SAS, ASA, AAS, or SSA).
• Solving Engine: Based on the detected case, it applies the appropriate mathematical laws:
  • Law of Cosines: Primarily used in SSS (to find angles) and SAS (to find the third side). Formula: c² = a² + b² - 2ab cos(C) (and its variations).
  • Law of Sines: Primarily used in ASA, AAS, and SSA cases (to find missing sides or angles). Formula: a/sin(A) = b/sin(B) = c/sin(C).
  • Sum of Angles: The fact that the three interior angles of any triangle always add up to 180° (or π radians) is used to find the final angle once two are known.
• Triangle Inequality Check: After attempting to solve for all sides (especially in angle-based inputs), it re-validates using the Triangle Inequality Theorem (sum of any two sides > third side).
• Additional Calculations: Once all sides and angles are determined, it calculates:
  • Perimeter: a + b + c
  • Area: Using Heron's formula (based on sides) or 0.5 * a * b * sin(C).
  • Type Classification: Determines if the triangle is Equilateral, Isosceles, or Scalene (by sides) and Acute, Right, or Obtuse (by angles).
• Display: Presents all calculated values (sides, angles in the selected unit, perimeter, area, type) in the results section.

Frequently Asked Questions (FAQs)

• How many values do I need to enter?
  You must enter exactly three values, and at least one of them must be a side length.
• What are the valid combinations (cases)?
  This calculator handles SSS (Side-Side-Side), SAS (Side-Angle-Side), ASA (Angle-Side-Angle), and AAS (Angle-Angle-Side). It also attempts to solve SSA (Side-Side-Angle), but be aware this case can sometimes have two possible solutions or no solution; the calculator typically finds one valid solution if it exists. AAA (Angle-Angle-Angle) is not enough to determine a unique triangle's size.
• What units should I use?
  Enter side lengths in any consistent unit (cm, inches, etc.). The calculated sides, perimeter, and area will be in those same units (or units squared for area). Ensure you select the correct unit (Degrees or Radians) for any angles you enter or expect as output.
• What is the difference between Degrees and Radians?
  They are two different units for measuring angles. A full circle is 360° or 2π radians. Most everyday geometry uses degrees, while radians are common in higher mathematics and physics. Ensure your input matches the selected unit. (e.g., 90 degrees = pi/2 radians ≈ 1.571 radians).
• Why did I get an "Invalid Input" or "Impossible Triangle" error?
  This could happen if:
  • You didn't enter exactly 3 values, or didn't include a side.
  • Entered values are non-positive (sides must be > 0, angles > 0).
  • The sum of entered angles is ≥ 180° (or π radians).
  • The provided values violate the Law of Sines (e.g., sin(A) > 1).
  • The provided sides violate the Triangle Inequality Theorem (sum of two sides must be > third side).
• How accurate are the results?
  Calculations use standard floating-point arithmetic. Results are generally very accurate for typical use, rounded to a few decimal places. Minor rounding differences may occur compared to manual calculations or other calculators.
• Is this Triangle Calculator free?
  Yes, this tool is completely free.