Triangle Calculator

How to Use This Tool

Follow these steps to calculate triangle properties:

1. Select a case from the dropdown that matches your known values (SSS, SAS, ASA, AAS, or SSA).
2. Enter the known measurements in the input fields that appear. Use the angle unit selector to choose degrees or radians.
3. Click "Calculate" to compute all unknown sides, angles, area, and perimeter. For SSA cases, the tool will show both possible triangles if they exist.
4. Use "Reset" to clear all inputs and start over. Results can be copied to your clipboard with the "Copy Results" button.

Formula and Logic

This calculator implements standard geometric and trigonometric laws:

• Law of Sines: a/sin(A) = b/sin(B) = c/sin(C) = 2R (circumdiameter).
• Law of Cosines: c² = a² + b² - 2ab·cos(C) (and cyclic permutations).
• Heron's Formula: Area = √[s(s-a)(s-b)(s-c)] where s = (a+b+c)/2.
• Area (SAS): ½·a·b·sin(C).
• Angle Sum: A + B + C = π radians (180°).

For SSA (ambiguous case), the calculator checks the condition a < b·sin(A) (no solution), a = b·sin(A) (one right triangle), and a > b·sin(A) with A acute (two solutions).

Practical Notes

• Precision & Rounding: Internal calculations use double-precision floating point. Displayed values are rounded to 6 decimal places. For engineering work, round final results to appropriate significant figures based on input precision.
• Input Validation: All sides must be positive. Angles must be >0 and <180° (or <π rad). For SSS, triangle inequality must hold (sum of any two sides > third side). For ASA/AAS, sum of given angles must be <180°.
• Edge Cases:
  • Degenerate triangles (e.g., collinear points) are rejected.
  • SSA with A ≥ 90° and a ≤ b yields no solution.
  • Right triangles (one angle = 90°) are handled naturally in SAS/SSS cases.
• Units: Side lengths can be any unit (meters, feet, etc.) as long as consistent. Angles can be degrees or radians—just ensure all angle inputs use the same unit.

Why This Tool Is Useful

Triangles are foundational in geometry, civil engineering, architecture, navigation, and physics. This calculator eliminates tedious manual computation, reduces errors, and helps visualize triangle relationships. It's ideal for trigonometry students verifying homework, engineers checking survey data, and designers solving spatial problems quickly.

Frequently Asked Questions

What is the "ambiguous case" in SSA and when does it occur?

The ambiguous case (SSA) occurs when given two sides and a non-included angle. It can produce two distinct triangles, one triangle, or no triangle. Two triangles exist when the given angle is acute, the side opposite the angle is shorter than the other given side but longer than the altitude (a < b and a > b·sin(A)). The calculator automatically detects and displays both solutions when applicable.

Can I use this for right-angled triangles?

Yes. For right triangles, use SAS with the right angle as the included angle, or SSS if you know all three sides. The calculator will correctly compute the missing sides/angles, including the 90° (or π/2 rad) angle. You can also use SSA if you know the hypotenuse and one leg plus the right angle (though SAS is simpler).

Why does my SSA calculation show "No solution"?

SSA has no solution when the side opposite the given angle is too short to reach the other side (a < b·sin(A)). This means the given angle is too small relative to the sides to form a closed triangle. Also, if the given angle is obtuse (≥90°) and the opposite side is not the longest (a ≤ b), no triangle exists. Double-check your inputs and try swapping which side/angle you consider as "a" and "A" (standard notation: side a opposite angle A).

Additional Guidance

When entering values, consider significant figures. The calculator does not enforce sig-fig rules—it computes exact values from your inputs. For real-world measurements, round results to match the least precise input. If you get unexpected results, verify that your angle unit (degrees/radians) matches your input values. Remember that in standard triangle notation, side a is opposite angle A, side b opposite B, and side c opposite C—this convention is used in all case definitions.

For educational purposes, manually verify a few results using the laws of sines/cosines to build intuition. The calculator is a verification tool, not a substitute for understanding the underlying geometry.