Triangle solver for SSS SAS ASA and AAS

Opposite side labels determine which rule applies

The side labels a, b and c are opposite angles A, B and C. They do not mean smallest, middle and largest unless the corresponding angles happen to have that order. The form names the actual measurements for every configuration, so the same letter does not silently switch meaning when you change methods. All side lengths must use one common unit.

In SSS mode, all three sides are known and the triangle must satisfy strict triangle inequalities. For example, 3, 4 and 5 work, but 1, 1 and 2 describe a flat boundary rather than a triangle with positive area. Rejecting that boundary prevents a zero-area shape from generating undefined radii or a misleading plot.

The trigonometric function grapher instead displays complete transformed sine and cosine waves, their period, phase and vertical range. Triangle ratios and a global wave graph answer different questions.

Included and opposite measurements are different inputs

In SAS mode the known sides are b and c, with included angle A between them. The cosine rule determines a. In ASA mode the known angles are A and B, with c connecting their vertices, so c is opposite the derived C. In AAS mode the known side is instead a, opposite the already known A. Using the same numeric side in those last two modes generally produces a differently scaled triangle.

The third angle is 180° minus the two supplied angles. It must be positive, and the form enforces a small positive supported minimum. Once one opposite side-angle pair is known, the sine rule scales the other sides. This page implements those four unambiguous configurations only. It does not select one branch of an SSA problem and hide a possible second triangle.

Read the sketch alongside the solved table

The equal-scale SVG sketch places A at the origin, B on the positive horizontal axis and C above that axis. Reflection would produce the same side lengths, opposite angles, area and radii. The table supplies the numerical values, while the sketch helps you inspect the correspondence between the entered measurements and the resulting shape. Very narrow triangles may appear flat at ordinary screen resolution.

For the default 3–4–5 example, area is 6 and perimeter is 12. The inradius equals twice the area divided by the perimeter, giving 1. The circumradius equals abc divided by four times the area, giving 2.5. The triangle area calculator can answer a simpler base-and-height question when those are the actual known measurements, rather than using a more elaborate solver to infer them. Trigonometric outputs are numerical approximations, not an uncertainty analysis of a measured drawing.