Triangle Properties:
Ambiguous Case (SSA) – Second Valid Triangle
Two distinct triangles can satisfy the given SSA measurements. The second solution is listed below.
How to Use This Calculator
- Select a Method: Choose ASA/AAS, SSA, SAS, or SSS.
- Enter Known Values: Sides must be positive. Angles in degrees (0 < angle < 180).
- Calculate: Click “Calculate Triangle”.
- Review Results: Sides, angles (deg/rad), perimeter, semiperimeter, area, inradius, circumradius, heights, and Law of Sines ratios appear. SSA may yield two solutions.
- Clear: Use “Clear Inputs & Results” to reset.
Reminder: Side a ↔ Angle A, b ↔ B, c ↔ C.
Mastering the Law of Sines: A Complete Guide
Introduction
The Law of Sines provides a proportional bridge between angles and sides in any oblique triangle. Where right–triangle methods fail, this identity succeeds. Its universal form—a/sin(A) = b/sin(B) = c/sin(C)—enables reconstruction from partial geometric data across surveying, navigation, astronomy, and structural analysis.
Conceptual Core
Each side “scaled” by the sine of its opposite angle yields one constant. Geometrically this constant equals the diameter of the circumcircle: a / sin(A) = 2R. Knowing any angle–side pair instantly defines the circumcircle, tightly constraining the rest of the triangle.
When the Law of Sines Excels
- ASA / AAS: Direct and unambiguous.
- SSA: Useful but can be ambiguous (0, 1, or 2 solutions).
- SAS / SSS: Typically start with Law of Cosines, then pivot to Law of Sines.
Ambiguous SSA Case
If given sides a, b and angle A opposite a, side a may “swing” to create two feasible locations for the third vertex—producing two distinct triangles with different angle B (acute vs. obtuse). Threshold comparison uses the projected altitude b * sin(A).
Computed Properties
- Area: Heron’s formula & trigonometric form
(1/2)ab sin(C) - R (Circumradius):
a / (2 sin A) - r (Inradius):
Area / s, wheresis semiperimeter - Heights:
h_a = 2K/a, etc. - Law Ratio Check:
a/sin(A),b/sin(B),c/sin(C)
Conclusion
Mastery comes from recognizing when proportional symmetry solves the system. Cross-verifying angles, side inequalities, and multiple area forms preserves numerical confidence. The Law of Sines remains a concise gateway to full triangle resolution.